(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 6.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[     69769,       2268]
NotebookOptionsPosition[     60433,       1950]
NotebookOutlinePosition[     62285,       2008]
CellTagsIndexPosition[     61863,       1995]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["RegularSequences 1.05", "Title"],

Cell["documentation notebook", "Subtitle"],

Cell[TextData[{
 "Eric Rowland\n",
 ButtonBox["http://math.tulane.edu/~erowland/packages.html",
  BaseStyle->"Hyperlink",
  ButtonData:>{
    URL["http://math.tulane.edu/~erowland/packages.html"], None}]
}], "Subsubtitle"],

Cell[CellGroupData[{

Cell["Introduction", "Section"],

Cell["\<\
RegularSequences is a package for identifying and analyzing automatic and \
regular sequences.
This introduction gives an overview of the main features of the package; the \
next section provides a complete list of package symbols along with their \
usage messages and further examples.\
\>", "Text"],

Cell[TextData[{
 "To use RegularSequences, first you will need to load the package by \
evaluating the following cell.  ",
 "(If you need help, see ",
 ButtonBox["loading a package",
  BaseStyle->"Hyperlink",
  ButtonData->{
    URL["http://math.tulane.edu/~erowland/packages/loadingapackage.html"], 
    None},
  ButtonNote->
   "http://math.tulane.edu/~erowland/packages/loadingapackage.html"],
 ".)"
}], "Text"],

Cell[BoxData[
 RowBox[{"<<", "RegularSequences`"}]], "Code"],

Cell[CellGroupData[{

Cell["Automatic and regular sequences", "Subsection"],

Cell[TextData[{
 "A sequence ",
 Cell[BoxData[
  FormBox[
   RowBox[{"a", "(", "n", ")"}], TraditionalForm]]],
 " is ",
 Cell[BoxData[
  FormBox["k", TraditionalForm]]],
 "-automatic if there is a finite automaton that outputs ",
 Cell[BoxData[
  FormBox[
   RowBox[{"a", "(", "n", ")"}], TraditionalForm]]],
 " when fed the base-",
 Cell[BoxData[
  FormBox["k", TraditionalForm]]],
 " digits of ",
 Cell[BoxData[
  FormBox["n", TraditionalForm]]],
 ".  Regular sequences are a natural generalization of automatic sequences.  \
They were introduced by Allouche and Shallit in the paper \"The ring of ",
 Cell[BoxData[
  FormBox["k", TraditionalForm]],
  FontSlant->"Plain"],
 "-regular sequences\" (",
 StyleBox["Theoretical Computer Science",
  FontSlant->"Italic"],
 " ",
 StyleBox["98",
  FontWeight->"Bold"],
 " (1992) 163\[Dash]197); the paper can be found ",
 ButtonBox["here",
  BaseStyle->"Hyperlink",
  ButtonData->{
    URL["http://www.cs.uwaterloo.ca/~shallit/Papers/as0.ps"], None},
  ButtonNote->"http://www.cs.uwaterloo.ca/~shallit/Papers/as0.ps"],
 "."
}], "Text"],

Cell[TextData[{
 "Let ",
 Cell[BoxData[
  FormBox[
   RowBox[{"a", "(", "n", ")"}], TraditionalForm]],
  FormatType->"TraditionalForm"],
 " be the parity of the number of 1s in the binary representation of ",
 Cell[BoxData[
  FormBox["n", TraditionalForm]],
  FormatType->"TraditionalForm"],
 "."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Table", "[", 
  RowBox[{
   RowBox[{"Mod", "[", 
    RowBox[{
     RowBox[{"DigitCount", "[", 
      RowBox[{"n", ",", "2", ",", "1"}], "]"}], ",", "2"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"n", ",", "0", ",", "31"}], "}"}]}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
  "0", ",", "1", ",", "1", ",", "0", ",", "1", ",", "0", ",", "0", ",", "1", 
   ",", "1", ",", "0", ",", "0", ",", "1", ",", "0", ",", "1", ",", "1", ",", 
   "0", ",", "1", ",", "0", ",", "0", ",", "1", ",", "0", ",", "1", ",", "1", 
   ",", "0", ",", "0", ",", "1", ",", "1", ",", "0", ",", "1", ",", "0", ",", 
   "0", ",", "1"}], "}"}]], "Output"]
}, Open  ]],

Cell[TextData[{
 "That sequence is 2-automatic.  Therefore it is also 2-regular, and ",
 StyleBox["FindRegularSequenceFunction", "Input"],
 " gives a representation for this sequence."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"FindRegularSequenceFunction", "[", 
  RowBox[{"%", ",", "2"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"1", ",", "0"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"1", ",", "0"}], "}"}]}], "}"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "1"}], "}"}]}], "]"}]], "Output"]
}, Open  ]],

Cell[TextData[{
 StyleBox["FindRegularSequenceFunction", "Input"],
 " returns a ",
 StyleBox["RegularSequence", "Input"],
 " object, which behaves as a function like ",
 StyleBox["DifferenceRoot", "Input"],
 "."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"%", "[", 
  RowBox[{"Range", "[", 
   RowBox[{"0", ",", "31"}], "]"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
  "0", ",", "1", ",", "1", ",", "0", ",", "1", ",", "0", ",", "0", ",", "1", 
   ",", "1", ",", "0", ",", "0", ",", "1", ",", "0", ",", "1", ",", "1", ",", 
   "0", ",", "1", ",", "0", ",", "0", ",", "1", ",", "0", ",", "1", ",", "1", 
   ",", "0", ",", "0", ",", "1", ",", "1", ",", "0", ",", "1", ",", "0", ",", 
   "0", ",", "1"}], "}"}]], "Output"]
}, Open  ]],

Cell[TextData[{
 "If we consider instead the total number of 1s in the binary representation \
of ",
 Cell[BoxData[
  FormBox["n", TraditionalForm]]],
 ", this sequence does not take finitely many values, so it is not automatic."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Table", "[", 
  RowBox[{
   RowBox[{"DigitCount", "[", 
    RowBox[{"n", ",", "2", ",", "1"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"n", ",", "0", ",", "31"}], "}"}]}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
  "0", ",", "1", ",", "1", ",", "2", ",", "1", ",", "2", ",", "2", ",", "3", 
   ",", "1", ",", "2", ",", "2", ",", "3", ",", "2", ",", "3", ",", "3", ",", 
   "4", ",", "1", ",", "2", ",", "2", ",", "3", ",", "2", ",", "3", ",", "3", 
   ",", "4", ",", "2", ",", "3", ",", "3", ",", "4", ",", "3", ",", "4", ",", 
   "4", ",", "5"}], "}"}]], "Output"]
}, Open  ]],

Cell["However, it is 2-regular:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"FindRegularSequenceFunction", "[", 
  RowBox[{"%", ",", "2"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"1", ",", "0"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{
         RowBox[{"-", "1"}], ",", "2"}], "}"}]}], "}"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "1"}], "}"}]}], "]"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"%", "[", 
  RowBox[{"Range", "[", 
   RowBox[{"0", ",", "31"}], "]"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
  "0", ",", "1", ",", "1", ",", "2", ",", "1", ",", "2", ",", "2", ",", "3", 
   ",", "1", ",", "2", ",", "2", ",", "3", ",", "2", ",", "3", ",", "3", ",", 
   "4", ",", "1", ",", "2", ",", "2", ",", "3", ",", "2", ",", "3", ",", "3", 
   ",", "4", ",", "2", ",", "3", ",", "3", ",", "4", ",", "3", ",", "4", ",", 
   "4", ",", "5"}], "}"}]], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Studying a regular sequence", "Subsection"],

Cell["\<\
The Thue\[Dash]Morse sequence is a famous example of a 2-automatic sequence.  \
It is an infinite sequence of 0s and 1s with the property that no block of \
entries ever occurs three times in a row.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"ThueMorse", "/@", 
  RowBox[{"Range", "[", 
   RowBox[{"0", ",", "31"}], "]"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
  "0", ",", "1", ",", "1", ",", "0", ",", "1", ",", "0", ",", "0", ",", "1", 
   ",", "1", ",", "0", ",", "0", ",", "1", ",", "0", ",", "1", ",", "1", ",", 
   "0", ",", "1", ",", "0", ",", "0", ",", "1", ",", "0", ",", "1", ",", "1", 
   ",", "0", ",", "0", ",", "1", ",", "1", ",", "0", ",", "1", ",", "0", ",", 
   "0", ",", "1"}], "}"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"tm", "=", 
  RowBox[{"FindRegularSequenceFunction", "[", 
   RowBox[{"%", ",", "2"}], "]"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"1", ",", "0"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"1", ",", "0"}], "}"}]}], "}"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "1"}], "}"}]}], "]"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"tm", "[", 
  RowBox[{"Range", "[", 
   RowBox[{"0", ",", "31"}], "]"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
  "0", ",", "1", ",", "1", ",", "0", ",", "1", ",", "0", ",", "0", ",", "1", 
   ",", "1", ",", "0", ",", "0", ",", "1", ",", "0", ",", "1", ",", "1", ",", 
   "0", ",", "1", ",", "0", ",", "0", ",", "1", ",", "0", ",", "1", ",", "1", 
   ",", "0", ",", "0", ",", "1", ",", "1", ",", "0", ",", "1", ",", "0", ",", 
   "0", ",", "1"}], "}"}]], "Output"]
}, Open  ]],

Cell[TextData[{
 "The Thue\[Dash]Morse sequence ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    SubscriptBox["t", "1"], "(", "n", ")"}], TraditionalForm]]],
 " satisfies the following system of linear recurrences."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceRecurrence", "[", 
  RowBox[{"tm", ",", "t", ",", "n"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{
       SubscriptBox["t", "1"], "[", 
       RowBox[{"2", " ", "n"}], "]"}], "\[Equal]", 
      RowBox[{
       SubscriptBox["t", "1"], "[", "n", "]"}]}], ",", 
     RowBox[{
      RowBox[{
       SubscriptBox["t", "1"], "[", 
       RowBox[{"1", "+", 
        RowBox[{"2", " ", "n"}]}], "]"}], "\[Equal]", 
      RowBox[{
       SubscriptBox["t", "2"], "[", "n", "]"}]}], ",", 
     RowBox[{
      RowBox[{
       SubscriptBox["t", "1"], "[", "0", "]"}], "\[Equal]", "0"}]}], "}"}], 
   ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{
       SubscriptBox["t", "2"], "[", 
       RowBox[{"2", " ", "n"}], "]"}], "\[Equal]", 
      RowBox[{
       SubscriptBox["t", "2"], "[", "n", "]"}]}], ",", 
     RowBox[{
      RowBox[{
       SubscriptBox["t", "2"], "[", 
       RowBox[{"1", "+", 
        RowBox[{"2", " ", "n"}]}], "]"}], "\[Equal]", 
      RowBox[{
       SubscriptBox["t", "1"], "[", "n", "]"}]}], ",", 
     RowBox[{
      RowBox[{
       SubscriptBox["t", "2"], "[", "0", "]"}], "\[Equal]", "1"}]}], "}"}]}], 
  "}"}]], "Output"]
}, Open  ]],

Cell[TextData[{
 "The two sequences ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    SubscriptBox["t", "1"], "(", "n", ")"}], TraditionalForm]]],
 " and ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    SubscriptBox["t", "2"], "(", "n", ")"}], TraditionalForm]]],
 " from that system are subsequences of the Thue\[Dash]Morse sequence:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"FindRegularSequenceRecurrence", "[", 
  RowBox[{
   RowBox[{"ThueMorse", "[", 
    RowBox[{"Range", "[", 
     RowBox[{"0", ",", "31"}], "]"}], "]"}], ",", "2", ",", 
   RowBox[{"t", "[", "n", "]"}]}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{"t", "[", 
       RowBox[{"2", " ", "n"}], "]"}], "\[Equal]", 
      RowBox[{"t", "[", "n", "]"}]}], ",", "True", ",", 
     RowBox[{
      RowBox[{"t", "[", "0", "]"}], "\[Equal]", "0"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{"t", "[", 
       RowBox[{"1", "+", 
        RowBox[{"4", " ", "n"}]}], "]"}], "\[Equal]", 
      RowBox[{"t", "[", 
       RowBox[{"1", "+", 
        RowBox[{"2", " ", "n"}]}], "]"}]}], ",", 
     RowBox[{
      RowBox[{"t", "[", 
       RowBox[{"1", "+", 
        RowBox[{"2", " ", 
         RowBox[{"(", 
          RowBox[{"1", "+", 
           RowBox[{"2", " ", "n"}]}], ")"}]}]}], "]"}], "\[Equal]", 
      RowBox[{"t", "[", "n", "]"}]}], ",", 
     RowBox[{
      RowBox[{"t", "[", "1", "]"}], "\[Equal]", "1"}]}], "}"}]}], 
  "}"}]], "Output"]
}, Open  ]],

Cell["Here are the first few terms:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceGeneratorTable", "[", 
  RowBox[{"tm", ",", "15"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
    "0", ",", "1", ",", "1", ",", "0", ",", "1", ",", "0", ",", "0", ",", "1",
      ",", "1", ",", "0", ",", "0", ",", "1", ",", "0", ",", "1", ",", "1", 
     ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
    "1", ",", "0", ",", "0", ",", "1", ",", "0", ",", "1", ",", "1", ",", "0",
      ",", "0", ",", "1", ",", "1", ",", "0", ",", "1", ",", "0", ",", "0", 
     ",", "1"}], "}"}]}], "}"}]], "Output"]
}, Open  ]],

Cell["\<\
The partial sums of the Thue\[Dash]Morse sequence grow linearly:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceExponent", "[", "tm", "]"}]], "Input"],

Cell[BoxData["1"], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"ListPlot", "[", 
  RowBox[{"Accumulate", "@", 
   RowBox[{"tm", "[", 
    RowBox[{"Range", "[", 
     RowBox[{"0", ",", "200"}], "]"}], "]"}]}], "]"}]], "Input"],

Cell[BoxData[
 GraphicsBox[
  {Hue[0.67, 0.6, 0.6], PointBox[CompressedData["
1:eJxd1UluFEEQheEQK5YsWLBgkSCEEEKIeQYHZp7BzKMLgxlWXCGOUkfJLbeo
I3AEjPhfS/laskJfp9VVlRnx6sDWr43tXRHxe+fvX/3/+bMW4yfH73fn+P0e
894c/2+feb+52e8ctN85ZD5sPmLXOWo+Ztc9btc9YT5p93HKfNp8xu7zrPmc
3fd5u+8L5ovmS1Q912WqnuuK+SpVz3nNvGZOKvtQsvbh+ugyxzqVfSpz3KCy
byVr326OLnPcorKvJWtfb48uWft8Z3SZ4y6Vcyhz3KNyLiXrXO6PLlnn9GB0
meMhlXMsWef4aHSZ4zGVcy5zPKFy7iXrnJ+OTqw+KHM3xzMqfZHmwuqbrvV1
6sboNBdWX3Wtq4+ej06sPitzN8cLKn2X5sLqy6519eHL0YnVp2Xu5nhFpW8T
q4/L3M3xmkpfp7mw+r5rXX3+ZnSaC2suutY1B29HJ9aclLmb4x2VuUmsOSpz
N8d7KnOV5sKau651zdmH0Yk1h2Xu5vhIZS7TXFhz27WuOf00Os2FNddd65rj
zdGJNedl7uaYqMx1w8qBNE/mwsqJ2dyxcmPR9ZQDn0c3c2LlyoRXuWKesXKm
mxdzbFHJlWZOrFya8CqXzDNWTnXzYo4vVHKpYeVYmidzYeXcbO5Yubfoesqx
r6ObObFyccKrXDTPWDnZzYs5tqnkYsPK0TRP5sLK2dncsXJ30fWUu99GN6wc
TvNkLqycns0dK7cXXU85/H10MydWrk94levmGSvnu3kxxw8qud7MifVemPDq
vWCesd4T3byY4yeV90LDO++RvyLWSVQ=
    "]]},
  AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
  Axes->True,
  AxesOrigin->{0, Automatic},
  PlotRange->Automatic,
  PlotRangeClipping->True]], "Output"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"Clear", "[", "tm", "]"}]], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Other examples of regular sequences", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceMatrixForm", "[", 
  RowBox[{"FindRegularSequenceFunction", "[", 
   RowBox[{
    RowBox[{"Accumulate", "[", 
     RowBox[{"ThueMorse", "[", 
      RowBox[{"Range", "[", 
       RowBox[{"0", ",", "31"}], "]"}], "]"}], "]"}], ",", "2"}], "]"}], 
  "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0", ",", "0", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "1", "0", "0"},
         {"0", "0", "0", "1"},
         {"0", 
          RowBox[{"-", "1"}], "1", "1"},
         {"0", 
          RowBox[{"-", "2"}], "0", "3"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]], ",", 
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "0", "1", "0"},
         {"0", 
          RowBox[{"-", "2"}], "2", "1"},
         {"0", "0", "2", "0"},
         {"0", 
          RowBox[{"-", "3"}], "3", "2"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "0", ",", "1", ",", "0"}], "}"}]}], "]"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceMatrixForm", "[", 
  RowBox[{"FindRegularSequenceFunction", "[", 
   RowBox[{
    RowBox[{"IntegerExponent", "[", 
     RowBox[{
      RowBox[{"Range", "[", "64", "]"}], ",", "2"}], "]"}], ",", "2"}], "]"}],
   "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "0"},
         {
          RowBox[{"-", "1"}], "1"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]], ",", 
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "1"},
         {
          RowBox[{"-", "1"}], "2"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "1"}], "}"}]}], "]"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceMatrixForm", "[", 
  RowBox[{"FindRegularSequenceFunction", "[", 
   RowBox[{
    RowBox[{"IntegerExponent", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"Range", "[", 
        RowBox[{"0", ",", "31"}], "]"}], "!"}], ",", "2"}], "]"}], ",", "2"}],
    "]"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "1", "0"},
         {
          RowBox[{"-", "2"}], "3", "0"},
         {
          RowBox[{"-", "4"}], "4", "1"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]], ",", 
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "1", "0"},
         {"0", "0", "1"},
         {"2", 
          RowBox[{"-", "5"}], "4"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "0", ",", "1"}], "}"}]}], "]"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceMatrixForm", "[", 
  RowBox[{"FindRegularSequenceFunction", "[", 
   RowBox[{
    RowBox[{"IntegerExponent", "[", 
     RowBox[{
      RowBox[{"Fibonacci", "[", 
       RowBox[{"Range", "[", 
        SuperscriptBox["2", "7"], "]"}], "]"}], ",", "2"}], "]"}], ",", "2"}],
    "]"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "1", "0", "0", "0"},
         {"0", "0", "0", "1", "0"},
         {"0", "3", "0", "0", "0"},
         {"0", "1", "0", "0", "0"},
         {"0", "0", "0", "0", "1"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]], ",", 
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "0", "1", "0", "0"},
         {"0", "0", "0", "0", "1"},
         {"0", "0", "1", "1", "0"},
         {"0", "0", "0", "1", "0"},
         {"0", "1", "0", "0", "0"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "0", ",", "0", ",", "0", ",", "1"}], "}"}]}], 
  "]"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceMatrixForm", "[", 
  RowBox[{"FindRegularSequenceFunction", "[", 
   RowBox[{
    RowBox[{"IntegerExponent", "[", 
     RowBox[{
      RowBox[{"Fibonacci", "[", 
       RowBox[{"Range", "[", "1000", "]"}], "]"}], ",", "4"}], "]"}], ",", 
    "4"}], "]"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "0", "0", "0", "0"},
         {"0", "1", "0", "0", "0"},
         {"0", "0", "0", "1", "0"},
         {"0", "0", "0", "1", "0"},
         {"0", "0", "0", "0", "1"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]], ",", 
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "1", "0", "0", "0"},
         {"0", "0", "0", "0", "1"},
         {"0", "2", "0", "0", "0"},
         {"0", "2", "0", "0", "0"},
         {"0", "0", "0", 
          FractionBox["1", "2"], "0"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]], ",", 
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "0", "0", "0", "0"},
         {"0", "0", "0", 
          FractionBox["1", "2"], "0"},
         {"0", "0", "0", "0", "2"},
         {"0", "0", "0", "0", "2"},
         {"0", "1", "0", "0", "0"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]], ",", 
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "0", "1", "0", "0"},
         {"0", "1", "0", "0", "0"},
         {"0", "0", "1", 
          FractionBox["1", "2"], "0"},
         {"0", "0", "0", "1", "0"},
         {"0", "0", "0", "0", "1"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "0", ",", "0", ",", "0", ",", "1"}], "}"}]}], 
  "]"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceMatrixForm", "[", 
  RowBox[{"FindRegularSequenceFunction", "[", 
   RowBox[{
    RowBox[{"Accumulate", "[", 
     RowBox[{"Table", "[", 
      RowBox[{
       RowBox[{"Boole", "[", 
        RowBox[{
         RowBox[{"Length", "[", 
          RowBox[{"PowersRepresentations", "[", 
           RowBox[{"n", ",", "3", ",", "2"}], "]"}], "]"}], ">", "0"}], "]"}],
        ",", 
       RowBox[{"{", 
        RowBox[{"n", ",", "0", ",", 
         SuperscriptBox["2", "6"]}], "}"}]}], "]"}], "]"}], ",", "2"}], "]"}],
   "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], 
   ",", 
   RowBox[{"{", 
    RowBox[{
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "1", "0", "0", "0", "0"},
         {"0", "0", "0", "1", "0", "0"},
         {"0", "0", "0", "0", "1", "0"},
         {
          RowBox[{"-", "2"}], "1", "0", "4", 
          RowBox[{"-", "3"}], "1"},
         {
          RowBox[{"-", "2"}], "1", "0", "3", 
          RowBox[{"-", "2"}], "1"},
         {
          RowBox[{"-", "2"}], "1", "0", "1", "0", "1"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]], ",", 
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "0", "1", "0", "0", "0"},
         {"0", "0", "0", 
          RowBox[{"-", "1"}], "2", "0"},
         {"0", "0", "0", "0", "0", "1"},
         {
          RowBox[{"-", "2"}], "0", "1", "1", "0", "1"},
         {
          RowBox[{"-", "2"}], "0", "1", "0", "1", "1"},
         {
          RowBox[{"-", "2"}], "0", "1", 
          RowBox[{"-", "1"}], "2", "1"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"1", ",", "1", ",", "2", ",", "1", ",", "2", ",", "4"}], "}"}]}], 
  "]"}]], "Output"]
}, Open  ]],

Cell[TextData[{
 "Counting the number of nonzero entries in the ",
 Cell[BoxData[
  FormBox["n", TraditionalForm]]],
 "th row of an additive cellular automaton produces a regular sequence:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceMatrixForm", "[", 
  RowBox[{"FindRegularSequenceFunction", "[", 
   RowBox[{
    RowBox[{"Total", "/@", 
     RowBox[{"CellularAutomaton", "[", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"27030", ",", "2", ",", 
         FractionBox["3", "2"]}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{
         RowBox[{"{", "1", "}"}], ",", "0"}], "}"}], ",", 
       RowBox[{
        SuperscriptBox["2", "5"], "-", "1"}]}], "]"}]}], ",", "2"}], "]"}], 
  "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"1", "0", "0"},
         {"4", "0", "0"},
         {"0", "1", "0"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]], ",", 
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "1", "0"},
         {"0", "0", "1"},
         {"0", "0", "2"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"1", ",", "4", ",", "4"}], "}"}]}], "]"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceMatrixForm", "[", 
  RowBox[{"FindRegularSequenceFunction", "[", 
   RowBox[{
    RowBox[{"Total", "/@", 
     RowBox[{"CellularAutomaton", "[", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"39270", ",", "2", ",", 
         FractionBox["3", "2"]}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{
         RowBox[{"{", "1", "}"}], ",", "0"}], "}"}], ",", 
       RowBox[{
        SuperscriptBox["2", "6"], "-", "1"}]}], "]"}]}], ",", "2"}], "]"}], 
  "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0", ",", "0", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"1", "0", "0", "0"},
         {"3", "0", "0", "0"},
         {"4", "1", "0", "0"},
         {"7", "2", "0", "0"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]], ",", 
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "1", "0", "0"},
         {"0", "0", "1", "0"},
         {"0", "0", "0", "1"},
         {
          RowBox[{"-", "2"}], "1", "0", "2"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"1", ",", "3", ",", "7", ",", "13"}], "}"}]}], "]"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceMatrixForm", "[", 
  RowBox[{"FindRegularSequenceFunction", "[", 
   RowBox[{
    RowBox[{"CellularAutomaton", "[", 
     RowBox[{"90", ",", 
      RowBox[{"{", 
       RowBox[{
        RowBox[{"{", "1", "}"}], ",", "0"}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{
        RowBox[{
         SuperscriptBox["2", "5"], "-", "1"}], ",", 
        RowBox[{"{", 
         RowBox[{"{", "3", "}"}], "}"}]}], "}"}]}], "]"}], ",", "2"}], "]"}], 
  "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0", ",", "0", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "0", "0", "0"},
         {"0", "0", "1", "0"},
         {"0", "0", "0", "0"},
         {"0", "0", 
          RowBox[{"-", "1"}], "1"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]], ",", 
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "1", "0", "0"},
         {"0", "0", "0", "1"},
         {"0", "0", "0", "1"},
         {"0", "0", "0", "1"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "0", ",", "0", ",", "1"}], "}"}]}], "]"}]], "Output"]
}, Open  ]]
}, Closed]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Package symbols", "Section"],

Cell[CellGroupData[{

Cell[TextData[StyleBox["FindRegularSequenceFunction", "Input"]], \
"Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"?", "FindRegularSequenceFunction"}]], "Input"],

Cell[BoxData[
 StyleBox["\<\"FindRegularSequenceFunction[{\!\(\*SubscriptBox[StyleBox[\\\"a\
\\\", \\\"TI\\\"], StyleBox[\\\"0\\\", \\\"TR\\\"]]\), \
\!\(\*SubscriptBox[StyleBox[\\\"a\\\", \\\"TI\\\"], StyleBox[\\\"1\\\", \
\\\"TR\\\"]]\), \*StyleBox[\\\"\[Ellipsis]\\\", \\\"TI\\\"]}, \
\*StyleBox[\\\"k\\\", \\\"TI\\\"]] attempts to find a simple RegularSequence \
function that yields the sequence \!\(\*SubscriptBox[StyleBox[\\\"a\\\", \
\\\"TI\\\"], StyleBox[\\\"n\\\", \\\"TI\\\"]]\) when given successive integer \
arguments.\\nFindRegularSequenceFunction[\*StyleBox[\\\"list\\\", \
\\\"TI\\\"], \*StyleBox[\\\"k\\\", \\\"TI\\\"], \*StyleBox[\\\"n\\\", \
\\\"TI\\\"]] gives the function applied to \*StyleBox[\\\"n\\\", \\\"TI\\\"].\
\"\>", "MSG"]], "Print", "PrintUsage",
 CellTags->"Info3480088158-6717758"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"FindRegularSequenceFunction", "[", 
  RowBox[{
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{"DigitCount", "[", 
      RowBox[{"n", ",", "2", ",", "1"}], "]"}], ",", 
     RowBox[{"{", 
      RowBox[{"n", ",", "0", ",", "31"}], "}"}]}], "]"}], ",", "2"}], 
  "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"1", ",", "0"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{
         RowBox[{"-", "1"}], ",", "2"}], "}"}]}], "}"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "1"}], "}"}]}], "]"}]], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["FindRegularSequenceRecurrence", "Input"]], \
"Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"?", 
  StyleBox["FindRegularSequenceRecurrence", "Input"]}]], "Input"],

Cell[BoxData[
 StyleBox["\<\"FindRegularSequenceRecurrence[{\!\(\*SubscriptBox[StyleBox[\\\"\
a\\\", \\\"TI\\\"], StyleBox[\\\"0\\\", \\\"TR\\\"]]\), \
\!\(\*SubscriptBox[StyleBox[\\\"a\\\", \\\"TI\\\"], StyleBox[\\\"1\\\", \
\\\"TR\\\"]]\), \*StyleBox[\\\"\[Ellipsis]\\\", \\\"TI\\\"]}, \
\*StyleBox[\\\"k\\\", \\\"TI\\\"], \*StyleBox[\\\"a\\\", \
\\\"TI\\\"][\*StyleBox[\\\"n\\\", \\\"TI\\\"]]] attempts to find linear \
relations between subsequences of \*StyleBox[\\\"a\\\", \
\\\"TI\\\"][\*StyleBox[\\\"n\\\", \\\"TI\\\"]] of the form \*StyleBox[\\\"a\\\
\", \\\"TI\\\"][\*StyleBox[\\\"k\\\", \\\"TI\\\"]^\*StyleBox[\\\"e\\\", \
\\\"TI\\\"] \*StyleBox[\\\"n\\\", \\\"TI\\\"] + \*StyleBox[\\\"i\\\", \
\\\"TI\\\"]].\"\>", "MSG"]], "Print", "PrintUsage",
 CellTags->"Info3480088161-5739138"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  StyleBox["FindRegularSequenceRecurrence", "Input"], "[", 
  RowBox[{
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{"DigitCount", "[", 
      RowBox[{"n", ",", "2", ",", "1"}], "]"}], ",", 
     RowBox[{"{", 
      RowBox[{"n", ",", "0", ",", "31"}], "}"}]}], "]"}], ",", "2", ",", 
   RowBox[{"a", "[", "n", "]"}]}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{"a", "[", 
       RowBox[{"2", " ", "n"}], "]"}], "\[Equal]", 
      RowBox[{"a", "[", "n", "]"}]}], ",", "True", ",", 
     RowBox[{
      RowBox[{"a", "[", "0", "]"}], "\[Equal]", "0"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{"a", "[", 
       RowBox[{"1", "+", 
        RowBox[{"4", " ", "n"}]}], "]"}], "\[Equal]", 
      RowBox[{"a", "[", 
       RowBox[{"1", "+", 
        RowBox[{"2", " ", "n"}]}], "]"}]}], ",", 
     RowBox[{
      RowBox[{"a", "[", 
       RowBox[{"1", "+", 
        RowBox[{"2", " ", 
         RowBox[{"(", 
          RowBox[{"1", "+", 
           RowBox[{"2", " ", "n"}]}], ")"}]}]}], "]"}], "\[Equal]", 
      RowBox[{
       RowBox[{"-", 
        RowBox[{"a", "[", "n", "]"}]}], "+", 
       RowBox[{"2", " ", 
        RowBox[{"a", "[", 
         RowBox[{"1", "+", 
          RowBox[{"2", " ", "n"}]}], "]"}]}]}]}], ",", 
     RowBox[{
      RowBox[{"a", "[", "1", "]"}], "\[Equal]", "1"}]}], "}"}]}], 
  "}"}]], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["RegularSequence", "Input"]], "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"?", "RegularSequence"}]], "Input"],

Cell[BoxData[
 StyleBox["\<\"RegularSequence[\*StyleBox[\\\"\[Lambda]\\\", \\\"TI\\\"], {\!\
\(\*SubscriptBox[StyleBox[\\\"m\\\", \\\"TI\\\"], StyleBox[\\\"0\\\", \
\\\"TR\\\"]]\), \!\(\*SubscriptBox[StyleBox[\\\"m\\\", \\\"TI\\\"], StyleBox[\
\\\"1\\\", \\\"TR\\\"]]\), \*StyleBox[\\\"\[Ellipsis]\\\", \\\"TI\\\"], \
\!\(\*SubscriptBox[StyleBox[\\\"m\\\", \\\"TI\\\"], StyleBox[\\\"k-1\\\", \
\\\"TI\\\"]]\)}, \*StyleBox[\\\"\[Kappa]\\\", \\\"TI\\\"]][\*StyleBox[\\\"n\\\
\", \\\"TI\\\"]] gives \*StyleBox[\\\"\[Lambda]\\\", \
\\\"TI\\\"].\!\(\*SubscriptBox[StyleBox[\\\"m\\\", \\\"TI\\\"], \
StyleBox[\\\"n0\\\", \\\"TI\\\"]]\).\!\(\*SubscriptBox[StyleBox[\\\"m\\\", \\\
\"TI\\\"], StyleBox[\\\"n1\\\", \
\\\"TI\\\"]]\).\[CenterEllipsis].\!\(\*SubscriptBox[StyleBox[\\\"m\\\", \
\\\"TI\\\"], StyleBox[\\\"nl\\\", \\\"TI\\\"]]\).\*StyleBox[\\\"\[Kappa]\\\", \
\\\"TI\\\"], where \*StyleBox[\\\"n\\\", \\\"TI\\\"] \[Equal] \
\!\(\*SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\\\"], StyleBox[\\\"l\\\", \
\\\"TI\\\"]]\)\[CenterEllipsis]\!\(\*SubscriptBox[StyleBox[\\\"n\\\", \
\\\"TI\\\"], StyleBox[\\\"1\\\", \\\"TR\\\"]]\)\!\(\*SubscriptBox[StyleBox[\\\
\"n\\\", \\\"TI\\\"], StyleBox[\\\"0\\\", \\\"TR\\\"]]\) in base \
\*StyleBox[\\\"k\\\", \\\"TI\\\"].\\nRegularSequence[\*StyleBox[\\\"\[Lambda]\
\\\", \\\"TI\\\"], \*StyleBox[\\\"matrices\\\", \\\"TI\\\"], \*StyleBox[\\\"\
\[Kappa]\\\", \\\"TI\\\"]][{\!\(\*SubscriptBox[StyleBox[\\\"n\\\", \
\\\"TI\\\"], StyleBox[\\\"1\\\", \\\"TR\\\"]]\), \!\(\*SubscriptBox[StyleBox[\
\\\"n\\\", \\\"TI\\\"], StyleBox[\\\"2\\\", \\\"TR\\\"]]\), \*StyleBox[\\\"\
\[Ellipsis]\\\", \\\"TI\\\"]}] gives terms \
\!\(\*SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\\\"], StyleBox[\\\"1\\\", \
\\\"TR\\\"]]\), \!\(\*SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\\\"], StyleBox[\
\\\"2\\\", \\\"TR\\\"]]\), \*StyleBox[\\\"\[Ellipsis]\\\", \\\"TI\\\"] of a \
\*StyleBox[\\\"k\\\", \\\"TI\\\"]\[Hyphen]regular sequence.\"\>", 
  "MSG"]], "Print", "PrintUsage",
 CellTags->"Info3480088163-1787245"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"RegularSequence", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"1", ",", "0"}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
        RowBox[{"{", 
         RowBox[{"1", ",", "0"}], "}"}], ",", 
        RowBox[{"{", 
         RowBox[{"0", ",", "1"}], "}"}]}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{
        RowBox[{"{", 
         RowBox[{"0", ",", "1"}], "}"}], ",", 
        RowBox[{"{", 
         RowBox[{
          RowBox[{"-", "1"}], ",", "2"}], "}"}]}], "}"}]}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{"0", ",", "1"}], "}"}]}], "]"}], "/@", 
  RowBox[{"Range", "[", 
   RowBox[{"0", ",", "31"}], "]"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
  "0", ",", "1", ",", "1", ",", "2", ",", "1", ",", "2", ",", "2", ",", "3", 
   ",", "1", ",", "2", ",", "2", ",", "3", ",", "2", ",", "3", ",", "3", ",", 
   "4", ",", "1", ",", "2", ",", "2", ",", "3", ",", "2", ",", "3", ",", "3", 
   ",", "4", ",", "2", ",", "3", ",", "3", ",", "4", ",", "3", ",", "4", ",", 
   "4", ",", "5"}], "}"}]], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["RegularSequenceExponent", "Input"]], "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"?", "RegularSequenceExponent"}]], "Input"],

Cell[BoxData[
 StyleBox["\<\"RegularSequenceExponent[RegularSequence[\*StyleBox[\\\"\
\[Lambda]\\\", \\\"TI\\\"], \*StyleBox[\\\"matrices\\\", \\\"TI\\\"], \
\*StyleBox[\\\"\[Kappa]\\\", \\\"TI\\\"]]] gives the exponent in the \
asymptotic growth rate of the partial sums of a \*StyleBox[\\\"k\\\", \
\\\"TI\\\"]\[Hyphen]regular sequence.\"\>", "MSG"]], "Print", "PrintUsage",
 CellTags->"Info3480088166-3369570"]
}, Open  ]],

Cell["Compute the fractal dimension of the Sierpinski sieve:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{"FindRegularSequenceFunction", "[", 
  RowBox[{
   RowBox[{"Total", "/@", 
    RowBox[{"CellularAutomaton", "[", 
     RowBox[{"90", ",", 
      RowBox[{"{", 
       RowBox[{
        RowBox[{"{", "1", "}"}], ",", "0"}], "}"}], ",", "100"}], "]"}]}], 
   ",", "2"}], "]"}], "\[IndentingNewLine]", 
 RowBox[{"RegularSequenceExponent", "[", "%", "]"}]}], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", "1", "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"{", "1", "}"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"{", "2", "}"}], "}"}]}], "}"}], ",", 
   RowBox[{"{", "1", "}"}]}], "]"}]], "Output"],

Cell[BoxData[
 FractionBox[
  RowBox[{"Log", "[", "3", "]"}], 
  RowBox[{"Log", "[", "2", "]"}]]], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["RegularSequenceGeneratorTable", "Input"]], \
"Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"?", "RegularSequenceGeneratorTable"}]], "Input"],

Cell[BoxData[
 StyleBox["\<\"RegularSequenceGeneratorTable[RegularSequence[\*StyleBox[\\\"\
\[Lambda]\\\", \\\"TI\\\"], \*StyleBox[\\\"matrices\\\", \\\"TI\\\"], \
\*StyleBox[\\\"\[Kappa]\\\", \\\"TI\\\"]], \*StyleBox[\\\"n\\\", \\\"TI\\\"]] \
gives terms 0 through \*StyleBox[\\\"n\\\", \\\"TI\\\"] of the generators of \
the additive group generated by the \*StyleBox[\\\"k\\\", \
\\\"TI\\\"]\[Hyphen]kernel of a \*StyleBox[\\\"k\\\", \
\\\"TI\\\"]\[Hyphen]regular sequence, where the first generator is the \
sequence itself.\"\>", "MSG"]], "Print", "PrintUsage",
 CellTags->"Info3480088168-3713130"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"FindRegularSequenceFunction", "[", 
  RowBox[{
   RowBox[{"DigitCount", "[", 
    RowBox[{
     RowBox[{"Range", "[", 
      RowBox[{"0", ",", "31"}], "]"}], ",", "2", ",", "1"}], "]"}], ",", 
   "2"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"1", ",", "0"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{
         RowBox[{"-", "1"}], ",", "2"}], "}"}]}], "}"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "1"}], "}"}]}], "]"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  StyleBox["RegularSequenceGeneratorTable", "Input"], "[", 
  RowBox[{"%", ",", "31"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
    "0", ",", "1", ",", "1", ",", "2", ",", "1", ",", "2", ",", "2", ",", "3",
      ",", "1", ",", "2", ",", "2", ",", "3", ",", "2", ",", "3", ",", "3", 
     ",", "4", ",", "1", ",", "2", ",", "2", ",", "3", ",", "2", ",", "3", 
     ",", "3", ",", "4", ",", "2", ",", "3", ",", "3", ",", "4", ",", "3", 
     ",", "4", ",", "4", ",", "5"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
    "1", ",", "2", ",", "2", ",", "3", ",", "2", ",", "3", ",", "3", ",", "4",
      ",", "2", ",", "3", ",", "3", ",", "4", ",", "3", ",", "4", ",", "4", 
     ",", "5", ",", "2", ",", "3", ",", "3", ",", "4", ",", "3", ",", "4", 
     ",", "4", ",", "5", ",", "3", ",", "4", ",", "4", ",", "5", ",", "4", 
     ",", "5", ",", "5", ",", "6"}], "}"}]}], "}"}]], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["RegularSequenceMatrixForm", "Input"]], "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"?", "RegularSequenceMatrixForm"}]], "Input"],

Cell[BoxData[
 StyleBox["\<\"RegularSequenceMatrixForm[RegularSequence[\*StyleBox[\\\"\
\[Lambda]\\\", \\\"TI\\\"], \*StyleBox[\\\"matrices\\\", \\\"TI\\\"], \
\*StyleBox[\\\"\[Kappa]\\\", \\\"TI\\\"]]] displays \*StyleBox[\\\"matrices\\\
\", \\\"TI\\\"] in MatrixForm.\"\>", "MSG"]], "Print", "PrintUsage",
 CellTags->"Info3480088171-5140446"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"FindRegularSequenceFunction", "[", 
  RowBox[{
   RowBox[{"DigitCount", "[", 
    RowBox[{
     RowBox[{"Range", "[", 
      RowBox[{"0", ",", "31"}], "]"}], ",", "2", ",", "1"}], "]"}], ",", 
   "2"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"1", ",", "0"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{
         RowBox[{"-", "1"}], ",", "2"}], "}"}]}], "}"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "1"}], "}"}]}], "]"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceMatrixForm", "[", "%", "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"1", "0"},
         {"0", "1"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]], ",", 
     TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
         {"0", "1"},
         {
          RowBox[{"-", "1"}], "2"}
        },
        GridBoxAlignment->{
         "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
          "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
        GridBoxSpacings->{"Columns" -> {
            Offset[0.27999999999999997`], {
             Offset[0.7]}, 
            Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
            Offset[0.2], {
             Offset[0.4]}, 
            Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}],
      Function[BoxForm`e$, 
       MatrixForm[BoxForm`e$]]]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "1"}], "}"}]}], "]"}]], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["RegularSequenceRank", "Input"]], "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"?", "RegularSequenceRank"}]], "Input"],

Cell[BoxData[
 StyleBox["\<\"RegularSequenceRank[RegularSequence[\*StyleBox[\\\"\[Lambda]\\\
\", \\\"TI\\\"], \*StyleBox[\\\"matrices\\\", \\\"TI\\\"], \*StyleBox[\\\"\
\[Kappa]\\\", \\\"TI\\\"]]] gives the size of the vectors and matrices.\"\>", 
  "MSG"]], "Print", "PrintUsage",
 CellTags->"Info3480088175-3638881"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"FindRegularSequenceFunction", "[", 
  RowBox[{
   RowBox[{"DigitCount", "[", 
    RowBox[{
     RowBox[{"Range", "[", 
      RowBox[{"0", ",", "31"}], "]"}], ",", "2", ",", "1"}], "]"}], ",", 
   "2"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"1", ",", "0"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{
         RowBox[{"-", "1"}], ",", "2"}], "}"}]}], "}"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "1"}], "}"}]}], "]"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceRank", "[", "%", "]"}]], "Input"],

Cell[BoxData["2"], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["RegularSequenceRecurrence", "Input"]], "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"?", "RegularSequenceRecurrence"}]], "Input"],

Cell[BoxData[
 StyleBox["\<\"RegularSequenceRecurrence[RegularSequence[\*StyleBox[\\\"\
\[Lambda]\\\", \\\"TI\\\"], \*StyleBox[\\\"matrices\\\", \\\"TI\\\"], \
\*StyleBox[\\\"\[Kappa]\\\", \\\"TI\\\"]], \*StyleBox[\\\"a\\\", \\\"TI\\\"], \
\*StyleBox[\\\"n\\\", \\\"TI\\\"]] gives the recurrence relations satisfied \
by generators \!\(\*SubscriptBox[StyleBox[\\\"a\\\", \\\"TI\\\"], StyleBox[\\\
\"i\\\", \\\"TI\\\"]]\)[\*StyleBox[\\\"n\\\", \\\"TI\\\"]] of the additive \
group generated by the \*StyleBox[\\\"k\\\", \\\"TI\\\"]\[Hyphen]kernel of a \
\*StyleBox[\\\"k\\\", \\\"TI\\\"]\[Hyphen]regular sequence, where \
\!\(\*SubscriptBox[StyleBox[\\\"a\\\", \\\"TI\\\"], StyleBox[\\\"1\\\", \
\\\"TR\\\"]]\)[\*StyleBox[\\\"n\\\", \\\"TI\\\"]] is the sequence \
itself.\"\>", "MSG"]], "Print", "PrintUsage",
 CellTags->"Info3480088178-6728371"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"FindRegularSequenceFunction", "[", 
  RowBox[{
   RowBox[{"DigitCount", "[", 
    RowBox[{
     RowBox[{"Range", "[", 
      RowBox[{"0", ",", "31"}], "]"}], ",", "2", ",", "1"}], "]"}], ",", 
   "2"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"RegularSequence", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"1", ",", "0"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"1", ",", "0"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{
         RowBox[{"-", "1"}], ",", "2"}], "}"}]}], "}"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"0", ",", "1"}], "}"}]}], "]"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"RegularSequenceRecurrence", "[", 
  RowBox[{"%", ",", "a", ",", "n"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{
       SubscriptBox["a", "1"], "[", 
       RowBox[{"2", " ", "n"}], "]"}], "\[Equal]", 
      RowBox[{
       SubscriptBox["a", "1"], "[", "n", "]"}]}], ",", 
     RowBox[{
      RowBox[{
       SubscriptBox["a", "1"], "[", 
       RowBox[{"1", "+", 
        RowBox[{"2", " ", "n"}]}], "]"}], "\[Equal]", 
      RowBox[{
       SubscriptBox["a", "2"], "[", "n", "]"}]}], ",", 
     RowBox[{
      RowBox[{
       SubscriptBox["a", "1"], "[", "0", "]"}], "\[Equal]", "0"}]}], "}"}], 
   ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{
       SubscriptBox["a", "2"], "[", 
       RowBox[{"2", " ", "n"}], "]"}], "\[Equal]", 
      RowBox[{
       SubscriptBox["a", "2"], "[", "n", "]"}]}], ",", 
     RowBox[{
      RowBox[{
       SubscriptBox["a", "2"], "[", 
       RowBox[{"1", "+", 
        RowBox[{"2", " ", "n"}]}], "]"}], "\[Equal]", 
      RowBox[{
       RowBox[{"-", 
        RowBox[{
         SubscriptBox["a", "1"], "[", "n", "]"}]}], "+", 
       RowBox[{"2", " ", 
        RowBox[{
         SubscriptBox["a", "2"], "[", "n", "]"}]}]}]}], ",", 
     RowBox[{
      RowBox[{
       SubscriptBox["a", "2"], "[", "0", "]"}], "\[Equal]", "1"}]}], "}"}]}], 
  "}"}]], "Output"]
}, Open  ]],

Cell[TextData[{
 StyleBox["FindRegularSequenceRecurrence", "Input"],
 " gives the relations in terms of subsequences of the original rather than \
as generators of the ",
 Cell[BoxData[
  FormBox["k", TraditionalForm]]],
 "-kernel:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"FindRegularSequenceRecurrence", "[", 
  RowBox[{
   RowBox[{"DigitCount", "[", 
    RowBox[{
     RowBox[{"Range", "[", 
      RowBox[{"0", ",", "31"}], "]"}], ",", "2", ",", "1"}], "]"}], ",", "2", 
   ",", 
   RowBox[{"a", "[", "n", "]"}]}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{"a", "[", 
       RowBox[{"2", " ", "n"}], "]"}], "\[Equal]", 
      RowBox[{"a", "[", "n", "]"}]}], ",", "True", ",", 
     RowBox[{
      RowBox[{"a", "[", "0", "]"}], "\[Equal]", "0"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{"a", "[", 
       RowBox[{"1", "+", 
        RowBox[{"4", " ", "n"}]}], "]"}], "\[Equal]", 
      RowBox[{"a", "[", 
       RowBox[{"1", "+", 
        RowBox[{"2", " ", "n"}]}], "]"}]}], ",", 
     RowBox[{
      RowBox[{"a", "[", 
       RowBox[{"1", "+", 
        RowBox[{"2", " ", 
         RowBox[{"(", 
          RowBox[{"1", "+", 
           RowBox[{"2", " ", "n"}]}], ")"}]}]}], "]"}], "\[Equal]", 
      RowBox[{
       RowBox[{"-", 
        RowBox[{"a", "[", "n", "]"}]}], "+", 
       RowBox[{"2", " ", 
        RowBox[{"a", "[", 
         RowBox[{"1", "+", 
          RowBox[{"2", " ", "n"}]}], "]"}]}]}]}], ",", 
     RowBox[{
      RowBox[{"a", "[", "1", "]"}], "\[Equal]", "1"}]}], "}"}]}], 
  "}"}]], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["ThueMorse", "Input"]], "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"?", "ThueMorse"}]], "Input"],

Cell[BoxData[
 StyleBox["\<\"ThueMorse[\*StyleBox[\\\"n\\\", \\\"TI\\\"]] gives the \
\*StyleBox[\\\"n\\\", \\\"TI\\\"]th term in the Thue-Morse sequence.\"\>", 
  "MSG"]], "Print", "PrintUsage",
 CellTags->"Info3480088183-7386208"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"ThueMorse", "[", 
  RowBox[{"Range", "[", 
   RowBox[{"0", ",", "31"}], "]"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
  "0", ",", "1", ",", "1", ",", "0", ",", "1", ",", "0", ",", "0", ",", "1", 
   ",", "1", ",", "0", ",", "0", ",", "1", ",", "0", ",", "1", ",", "1", ",", 
   "0", ",", "1", ",", "0", ",", "0", ",", "1", ",", "0", ",", "1", ",", "1", 
   ",", "0", ",", "0", ",", "1", ",", "1", ",", "0", ",", "1", ",", "0", ",", 
   "0", ",", "1"}], "}"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"Reverse", "[", "%", "]"}], "\[Equal]", 
  RowBox[{"1", "-", "%"}]}]], "Input"],

Cell[BoxData["True"], "Output"]
}, Open  ]]
}, Closed]]
}, Open  ]]
}, Open  ]]
},
AutoGeneratedPackage->None,
WindowSize->{1272, 717},
WindowMargins->{{Automatic, 0}, {Automatic, 0}},
ShowSelection->True,
CellLabelAutoDelete->True,
TrackCellChangeTimes->False,
Magnification->1.,
FrontEndVersion->"7.0 for Microsoft Windows (64-bit) (February 18, 2009)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{
 "Info3480088158-6717758"->{
  Cell[37555, 1215, 816, 12, 59, "Print",
   CellTags->"Info3480088158-6717758"]},
 "Info3480088161-5739138"->{
  Cell[39580, 1279, 794, 12, 42, "Print",
   CellTags->"Info3480088161-5739138"]},
 "Info3480088163-1787245"->{
  Cell[42068, 1357, 2012, 29, 60, "Print",
   CellTags->"Info3480088163-1787245"]},
 "Info3480088166-3369570"->{
  Cell[45440, 1437, 413, 6, 40, "Print",
   CellTags->"Info3480088166-3369570"]},
 "Info3480088168-3713130"->{
  Cell[47000, 1491, 603, 9, 58, "Print",
   CellTags->"Info3480088168-3713130"]},
 "Info3480088171-5140446"->{
  Cell[49715, 1574, 344, 5, 40, "Print",
   CellTags->"Info3480088171-5140446"]},
 "Info3480088175-3638881"->{
  Cell[52805, 1678, 318, 5, 40, "Print",
   CellTags->"Info3480088175-3638881"]},
 "Info3480088178-6728371"->{
  Cell[54378, 1739, 845, 12, 59, "Print",
   CellTags->"Info3480088178-6728371"]},
 "Info3480088183-7386208"->{
  Cell[59414, 1913, 232, 4, 40, "Print",
   CellTags->"Info3480088183-7386208"]}
 }
*)
(*CellTagsIndex
CellTagsIndex->{
 {"Info3480088158-6717758", 60865, 1965},
 {"Info3480088161-5739138", 60977, 1968},
 {"Info3480088163-1787245", 61089, 1971},
 {"Info3480088166-3369570", 61202, 1974},
 {"Info3480088168-3713130", 61313, 1977},
 {"Info3480088171-5140446", 61424, 1980},
 {"Info3480088175-3638881", 61535, 1983},
 {"Info3480088178-6728371", 61646, 1986},
 {"Info3480088183-7386208", 61758, 1989}
 }
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[567, 22, 38, 0, 83, "Title"],
Cell[608, 24, 42, 0, 49, "Subtitle"],
Cell[653, 26, 222, 6, 47, "Subsubtitle"],
Cell[CellGroupData[{
Cell[900, 36, 31, 0, 71, "Section"],
Cell[934, 38, 310, 6, 47, "Text"],
Cell[1247, 46, 414, 12, 29, "Text"],
Cell[1664, 60, 60, 1, 47, "Code"],
Cell[CellGroupData[{
Cell[1749, 65, 53, 0, 36, "Subsection"],
Cell[1805, 67, 1079, 36, 47, "Text"],
Cell[2887, 105, 308, 11, 29, "Text"],
Cell[CellGroupData[{
Cell[3220, 120, 274, 8, 31, "Input"],
Cell[3497, 130, 392, 7, 30, "Output"]
}, Open  ]],
Cell[3904, 140, 196, 4, 30, "Text"],
Cell[CellGroupData[{
Cell[4125, 148, 103, 2, 31, "Input"],
Cell[4231, 152, 583, 20, 30, "Output"]
}, Open  ]],
Cell[4829, 175, 223, 7, 30, "Text"],
Cell[CellGroupData[{
Cell[5077, 186, 111, 3, 31, "Input"],
Cell[5191, 191, 392, 7, 30, "Output"]
}, Open  ]],
Cell[5598, 201, 241, 6, 29, "Text"],
Cell[CellGroupData[{
Cell[5864, 211, 214, 6, 31, "Input"],
Cell[6081, 219, 392, 7, 30, "Output"]
}, Open  ]],
Cell[6488, 229, 41, 0, 29, "Text"],
Cell[CellGroupData[{
Cell[6554, 233, 103, 2, 31, "Input"],
Cell[6660, 237, 608, 21, 30, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[7305, 263, 111, 3, 31, "Input"],
Cell[7419, 268, 392, 7, 30, "Output"]
}, Open  ]]
}, Closed]],
Cell[CellGroupData[{
Cell[7860, 281, 49, 0, 28, "Subsection"],
Cell[7912, 283, 222, 4, 29, "Text"],
Cell[CellGroupData[{
Cell[8159, 291, 115, 3, 31, "Input"],
Cell[8277, 296, 392, 7, 30, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[8706, 308, 128, 3, 31, "Input"],
Cell[8837, 313, 583, 20, 30, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[9457, 338, 112, 3, 31, "Input"],
Cell[9572, 343, 392, 7, 30, "Output"]
}, Open  ]],
Cell[9979, 353, 223, 7, 29, "Text"],
Cell[CellGroupData[{
Cell[10227, 364, 112, 2, 31, "Input"],
Cell[10342, 368, 1156, 40, 30, "Output"]
}, Open  ]],
Cell[11513, 411, 338, 12, 29, "Text"],
Cell[CellGroupData[{
Cell[11876, 427, 242, 6, 31, "Input"],
Cell[12121, 435, 910, 30, 30, "Output"]
}, Open  ]],
Cell[13046, 468, 45, 0, 29, "Text"],
Cell[CellGroupData[{
Cell[13116, 472, 107, 2, 31, "Input"],
Cell[13226, 476, 485, 12, 30, "Output"]
}, Open  ]],
Cell[13726, 491, 88, 2, 29, "Text"],
Cell[CellGroupData[{
Cell[13839, 497, 77, 1, 31, "Input"],
Cell[13919, 500, 28, 0, 30, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[13984, 505, 185, 5, 31, "Input"],
Cell[14172, 512, 1083, 22, 239, "Output"]
}, Open  ]],
Cell[15270, 537, 59, 1, 31, "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell[15366, 543, 57, 0, 28, "Subsection"],
Cell[CellGroupData[{
Cell[15448, 547, 302, 8, 31, "Input"],
Cell[15753, 557, 1807, 50, 72, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[17597, 612, 266, 7, 31, "Input"],
Cell[17866, 621, 1551, 44, 43, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[19454, 670, 319, 9, 31, "Input"],
Cell[19776, 681, 1669, 47, 57, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[21482, 733, 337, 9, 33, "Input"],
Cell[21822, 744, 1838, 49, 87, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[23697, 798, 311, 8, 31, "Input"],
Cell[24011, 808, 3471, 92, 97, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[27519, 905, 570, 16, 33, "Input"],
Cell[28092, 923, 2254, 62, 102, "Output"]
}, Open  ]],
Cell[30361, 988, 201, 5, 29, "Text"],
Cell[CellGroupData[{
Cell[30587, 997, 517, 15, 47, "Input"],
Cell[31107, 1014, 1591, 44, 57, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[32735, 1063, 517, 15, 47, "Input"],
Cell[33255, 1080, 1730, 47, 72, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[35022, 1132, 500, 15, 33, "Input"],
Cell[35525, 1149, 1729, 47, 72, "Output"]
}, Open  ]]
}, Closed]]
}, Open  ]],
Cell[CellGroupData[{
Cell[37315, 1203, 34, 0, 71, "Section"],
Cell[CellGroupData[{
Cell[37374, 1207, 83, 1, 28, "Subsubsection"],
Cell[CellGroupData[{
Cell[37482, 1212, 70, 1, 31, "Input"],
Cell[37555, 1215, 816, 12, 59, "Print",
 CellTags->"Info3480088158-6717758"]
}, Open  ]],
Cell[CellGroupData[{
Cell[38408, 1232, 305, 9, 31, "Input"],
Cell[38716, 1243, 608, 21, 30, "Output"]
}, Open  ]]
}, Closed]],
Cell[CellGroupData[{
Cell[39373, 1270, 85, 1, 22, "Subsubsection"],
Cell[CellGroupData[{
Cell[39483, 1275, 94, 2, 31, "Input"],
Cell[39580, 1279, 794, 12, 42, "Print",
 CellTags->"Info3480088161-5739138"]
}, Open  ]],
Cell[CellGroupData[{
Cell[40411, 1296, 365, 10, 31, "Input"],
Cell[40779, 1308, 1085, 36, 30, "Output"]
}, Open  ]]
}, Closed]],
Cell[CellGroupData[{
Cell[41913, 1350, 69, 0, 22, "Subsubsection"],
Cell[CellGroupData[{
Cell[42007, 1354, 58, 1, 31, "Input"],
Cell[42068, 1357, 2012, 29, 60, "Print",
 CellTags->"Info3480088163-1787245"]
}, Open  ]],
Cell[CellGroupData[{
Cell[44117, 1391, 708, 24, 31, "Input"],
Cell[44828, 1417, 392, 7, 30, "Output"]
}, Open  ]]
}, Closed]],
Cell[CellGroupData[{
Cell[45269, 1430, 77, 0, 22, "Subsubsection"],
Cell[CellGroupData[{
Cell[45371, 1434, 66, 1, 31, "Input"],
Cell[45440, 1437, 413, 6, 40, "Print",
 CellTags->"Info3480088166-3369570"]
}, Open  ]],
Cell[45868, 1446, 70, 0, 29, "Text"],
Cell[CellGroupData[{
Cell[45963, 1450, 384, 10, 52, "Input"],
Cell[46350, 1462, 305, 10, 30, "Output"],
Cell[46658, 1474, 108, 3, 49, "Output"]
}, Open  ]]
}, Closed]],
Cell[CellGroupData[{
Cell[46815, 1483, 85, 1, 22, "Subsubsection"],
Cell[CellGroupData[{
Cell[46925, 1488, 72, 1, 31, "Input"],
Cell[47000, 1491, 603, 9, 58, "Print",
 CellTags->"Info3480088168-3713130"]
}, Open  ]],
Cell[CellGroupData[{
Cell[47640, 1505, 243, 7, 31, "Input"],
Cell[47886, 1514, 608, 21, 30, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[48531, 1540, 128, 3, 31, "Input"],
Cell[48662, 1545, 829, 16, 50, "Output"]
}, Open  ]]
}, Closed]],
Cell[CellGroupData[{
Cell[49540, 1567, 79, 0, 22, "Subsubsection"],
Cell[CellGroupData[{
Cell[49644, 1571, 68, 1, 31, "Input"],
Cell[49715, 1574, 344, 5, 40, "Print",
 CellTags->"Info3480088171-5140446"]
}, Open  ]],
Cell[CellGroupData[{
Cell[50096, 1584, 243, 7, 31, "Input"],
Cell[50342, 1593, 608, 21, 30, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[50987, 1619, 78, 1, 31, "Input"],
Cell[51068, 1622, 1525, 43, 43, "Output"]
}, Open  ]]
}, Closed]],
Cell[CellGroupData[{
Cell[52642, 1671, 73, 0, 22, "Subsubsection"],
Cell[CellGroupData[{
Cell[52740, 1675, 62, 1, 31, "Input"],
Cell[52805, 1678, 318, 5, 40, "Print",
 CellTags->"Info3480088175-3638881"]
}, Open  ]],
Cell[CellGroupData[{
Cell[53160, 1688, 243, 7, 31, "Input"],
Cell[53406, 1697, 608, 21, 30, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[54051, 1723, 72, 1, 31, "Input"],
Cell[54126, 1726, 28, 0, 30, "Output"]
}, Open  ]]
}, Closed]],
Cell[CellGroupData[{
Cell[54203, 1732, 79, 0, 22, "Subsubsection"],
Cell[CellGroupData[{
Cell[54307, 1736, 68, 1, 31, "Input"],
Cell[54378, 1739, 845, 12, 59, "Print",
 CellTags->"Info3480088178-6728371"]
}, Open  ]],
Cell[CellGroupData[{
Cell[55260, 1756, 243, 7, 31, "Input"],
Cell[55506, 1765, 608, 21, 30, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[56151, 1791, 111, 2, 31, "Input"],
Cell[56265, 1795, 1301, 45, 30, "Output"]
}, Open  ]],
Cell[57581, 1843, 244, 7, 30, "Text"],
Cell[CellGroupData[{
Cell[57850, 1854, 284, 8, 31, "Input"],
Cell[58137, 1864, 1085, 36, 30, "Output"]
}, Open  ]]
}, Closed]],
Cell[CellGroupData[{
Cell[59271, 1906, 63, 0, 22, "Subsubsection"],
Cell[CellGroupData[{
Cell[59359, 1910, 52, 1, 31, "Input"],
Cell[59414, 1913, 232, 4, 40, "Print",
 CellTags->"Info3480088183-7386208"]
}, Open  ]],
Cell[CellGroupData[{
Cell[59683, 1922, 119, 3, 31, "Input"],
Cell[59805, 1927, 392, 7, 30, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[60234, 1939, 113, 3, 31, "Input"],
Cell[60350, 1944, 31, 0, 30, "Output"]
}, Open  ]]
}, Closed]]
}, Open  ]]
}, Open  ]]
}
]
*)

(* End of internal cache information *)