URL: https://mathworld.wolfram.com/notebooks/IntegerSequences/BitLength.nb
(* Content-type: application/vnd.wolfram.mathematica *)
(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)
(* CreatedBy='Mathematica 9.0' *)
(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[ 157, 7]
NotebookDataLength[ 18599, 520]
NotebookOptionsPosition[ 16853, 451]
NotebookOutlinePosition[ 17207, 467]
CellTagsIndexPosition[ 17164, 464]
WindowFrame->Normal*)
(* Beginning of Notebook Content *)
Notebook[{
Cell[CellGroupData[{
Cell["Bit Length", "Title"],
Cell[CellGroupData[{
Cell["Author", "Subsection"],
Cell["\<\
Eric W. Weisstein
November 2, 2008\
\>", "Text"],
Cell[TextData[{
"This notebook downloaded from ",
ButtonBox["http://mathworld.wolfram.com/notebooks/IntegerSequences/BitLength.\
nb",
BaseStyle->"Hyperlink",
ButtonData:>{
URL["http://mathworld.wolfram.com/notebooks/IntegerSequences/BitLength.\
nb"], None}],
"."
}], "Text"],
Cell[TextData[{
"For more information, see Eric's ",
StyleBox["MathWorld",
FontSlant->"Italic"],
" entry ",
ButtonBox["http://mathworld.wolfram.com/BitLength.html",
BaseStyle->"Hyperlink",
ButtonData:>{
URL["http://mathworld.wolfram.com/BitLength.html"], None}],
"."
}], "Text"],
Cell["\<\
\[Copyright]2008 Wolfram Research, Inc. except for portions noted otherwise\
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Code", "Subsection"],
Cell[BoxData[{
RowBox[{
RowBox[{"MyBitLength", "[", "n_", "]"}], ":=",
RowBox[{
RowBox[{"Floor", "[",
RowBox[{"Log2", "[", "n", "]"}], "]"}], "+",
"1"}]}], "\[IndentingNewLine]",
RowBox[{"SetAttributes", "[",
RowBox[{"MyBitLength", ",", "Listable"}], "]"}]}], "Input"],
Cell[BoxData[{
RowBox[{
RowBox[{"MyBitLength2", "[", "n_", "]"}], ":=",
RowBox[{"Ceiling", "[",
RowBox[{"Log2", "[",
RowBox[{"n", "+", "1"}], "]"}], "]"}]}], "\[IndentingNewLine]",
RowBox[{"SetAttributes", "[",
RowBox[{"MyBitLength2", ",", "Listable"}], "]"}]}], "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell["Identities", "Section"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"FunctionExpand", "[",
RowBox[{
RowBox[{"BitLength", "[", "n", "]"}], ",",
RowBox[{
RowBox[{"n", "\[Element]", "Integers"}], "&&",
RowBox[{"n", ">", "0"}]}]}], "]"}]], "Input"],
Cell[BoxData[
RowBox[{"BitLength", "[", "n", "]"}]], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"FunctionExpand", "[",
RowBox[{
RowBox[{"IntegerLength", "[",
RowBox[{"n", ",", "2"}], "]"}], ",",
RowBox[{
RowBox[{"n", "\[Element]", "Integers"}], "&&",
RowBox[{"n", ">", "0"}]}]}], "]"}]], "Input"],
Cell[BoxData[
RowBox[{"IntegerLength", "[",
RowBox[{"n", ",", "2"}], "]"}]], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"FunctionExpand", "[",
RowBox[{
RowBox[{"IntegerLength", "[", "n", "]"}], ",",
RowBox[{
RowBox[{"n", "\[Element]", "Integers"}], "&&",
RowBox[{"n", ">", "0"}]}]}], "]"}]], "Input"],
Cell[BoxData[
RowBox[{"1", "+",
RowBox[{"Floor", "[",
FractionBox[
RowBox[{"Log", "[", "n", "]"}],
RowBox[{"Log", "[", "10", "]"}]], "]"}]}]], "Output"]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["Table", "Section"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"l1", "=",
RowBox[{"BitLength", "[",
RowBox[{"Range", "[", "100", "]"}], "]"}]}]], "Input"],
Cell[BoxData[
RowBox[{"{",
RowBox[{
"1", ",", "2", ",", "2", ",", "3", ",", "3", ",", "3", ",", "3", ",", "4",
",", "4", ",", "4", ",", "4", ",", "4", ",", "4", ",", "4", ",", "4", ",",
"5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5",
",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",",
"5", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6",
",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",",
"6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6",
",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",",
"6", ",", "6", ",", "6", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7",
",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",",
"7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7",
",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",",
"7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7",
",", "7", ",", "7"}], "}"}]], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"l2", "=",
RowBox[{"MyBitLength", "[",
RowBox[{"Range", "[", "100", "]"}], "]"}]}]], "Input"],
Cell[BoxData[
RowBox[{"{",
RowBox[{
"1", ",", "2", ",", "2", ",", "3", ",", "3", ",", "3", ",", "3", ",", "4",
",", "4", ",", "4", ",", "4", ",", "4", ",", "4", ",", "4", ",", "4", ",",
"5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5",
",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",",
"5", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6",
",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",",
"6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6",
",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",",
"6", ",", "6", ",", "6", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7",
",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",",
"7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7",
",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",",
"7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7",
",", "7", ",", "7"}], "}"}]], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"l3", "=",
RowBox[{"MyBitLength2", "[",
RowBox[{"Range", "[", "100", "]"}], "]"}]}]], "Input"],
Cell[BoxData[
RowBox[{"{",
RowBox[{
"1", ",", "2", ",", "2", ",", "3", ",", "3", ",", "3", ",", "3", ",", "4",
",", "4", ",", "4", ",", "4", ",", "4", ",", "4", ",", "4", ",", "4", ",",
"5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5",
",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",", "5", ",",
"5", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6",
",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",",
"6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6",
",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",", "6", ",",
"6", ",", "6", ",", "6", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7",
",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",",
"7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7",
",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",",
"7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7", ",", "7",
",", "7", ",", "7"}], "}"}]], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Union", "/@",
RowBox[{"{",
RowBox[{
RowBox[{"l1", "-", "l2"}], ",",
RowBox[{"l1", "-", "l3"}]}], "}"}]}]], "Input"],
Cell[BoxData[
RowBox[{"{",
RowBox[{
RowBox[{"{", "0", "}"}], ",",
RowBox[{"{", "0", "}"}]}], "}"}]], "Output"]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["Plots", "Section"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"GraphicsGrid", "[",
RowBox[{
RowBox[{"Partition", "[",
RowBox[{
RowBox[{"Table", "[",
RowBox[{
RowBox[{"ListPlot", "[",
RowBox[{
RowBox[{"BitLength", "[",
RowBox[{"Range", "[",
RowBox[{"2", "^", "n"}], "]"}], "]"}], ",", "\[IndentingNewLine]",
RowBox[{"Joined", "\[Rule]", "True"}], ",",
RowBox[{"PlotRange", "\[Rule]", "All"}], ",",
RowBox[{"PlotStyle", "\[Rule]", "Red"}], ",",
RowBox[{"AxesLabel", "\[Rule]",
RowBox[{"{",
RowBox[{
RowBox[{"HoldForm", "[", "n", "]"}], ",",
RowBox[{"BL", "[",
RowBox[{"HoldForm", "[", "n", "]"}], "]"}]}], "}"}]}]}], "]"}],
",",
RowBox[{"{",
RowBox[{"n", ",", "5", ",", "10"}], "}"}]}], "]"}], ",", "3"}], "]"}],
",",
RowBox[{"ImageSize", "\[Rule]", "400"}], ",",
RowBox[{"Spacings", "\[Rule]",
RowBox[{"{",
RowBox[{
RowBox[{"-", "40"}], ",", "0"}], "}"}]}]}], "]"}]], "Input"],
Cell[BoxData[
GraphicsBox[{{}, {{InsetBox[
GraphicsBox[
{RGBColor[1, 0, 0], LineBox[CompressedData["
1:eJxdzjsOQEAUQNGJSqlQTKFARERE/P/MrMESJGpbtwSExp3k5ea84mW87Vh3
Qwjh3vP0faf6V+h/TdjC3oYl7MAu7vhwAIdwBMdwAqdwhn/ncAGXcAXXcAO3
cAf38ACP8ATP8AIrWH+V+gLYqhr0
"]]},
AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->True,
AxesLabel->{
FormBox[
TagBox["n", HoldForm], TraditionalForm],
FormBox[
RowBox[{"BL", "(",
TagBox["n", HoldForm], ")"}], TraditionalForm]},
AxesOrigin->{0, Automatic},
PlotRangeClipping->True], {160, -111.24611797498106`},
ImageScaled[{0.5, 0.5}], {360, 222.49223594996212`}], InsetBox[
GraphicsBox[
{RGBColor[1, 0, 0], LineBox[CompressedData["
1:eJxd0ctKAmEAhuEPVy1buHDhwkJEIiQrzU46ph08R5cQuPYWuuQuodMs8hn4
+XjmheGHOXrfvW0rSYrv87N/z+dof3/zvz3Ah7yv4hqu4wbfOcZN3MJtfIJP
cQefce8uPscX+BL3cB9f4QG+xjf4Ft/hezzEI1yUW/6nD5wxHeeBjjOh40zp
OI90nCc6zjMd54WOM6PjzOk4CzrOko6zouOs6TgbOs5rufXiCxmHPxg=
"]]},
AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->True,
AxesLabel->{
FormBox[
TagBox["n", HoldForm], TraditionalForm],
FormBox[
RowBox[{"BL", "(",
TagBox["n", HoldForm], ")"}], TraditionalForm]},
AxesOrigin->{0, Automatic},
PlotRangeClipping->True], {480, -111.24611797498106`},
ImageScaled[{0.5, 0.5}], {360, 222.49223594996212`}], InsetBox[
GraphicsBox[
{RGBColor[1, 0, 0], LineBox[CompressedData["
1:eJxd08tK1AEYxuEXVy1buGjRwkRCRCJLrTxOZgfPZeaxUgTX3cJ3ad8leQka
/jc+A8PHMz/ePwPDvLj+d3gzlqTu3//vw+t2/fHN6PF9gp/y+Th+hp/jCZ4z
iafwSzyNZ/AsfoVf873n8Bv8Fs/jBbyI3+H3+ANewst4Ba/iNbyOR8MdfqfC
+UjH2aDjfKLjbNJxPtNxvtBxvtJxvtFxtug423ScHTrOLh1nj46zT8c5oON8
H+7wvxzhwo3zgz0u3DiH7HHhxvnJHhdunCP2uHDj/GKPCzfOMXtcuHFO2OPC
jXPKHhdunDP2uHDjnLPHhRvngj0u3Di/2ePCjfOHPS7cOH/Z48KNc8keF26c
q+FOjO4ASLiUTA==
"]]},
AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->True,
AxesLabel->{
FormBox[
TagBox["n", HoldForm], TraditionalForm],
FormBox[
RowBox[{"BL", "(",
TagBox["n", HoldForm], ")"}], TraditionalForm]},
AxesOrigin->{0, Automatic},
PlotRangeClipping->True], {800, -111.24611797498106`},
ImageScaled[{0.5, 0.5}], {360, 222.49223594996212`}]}, {InsetBox[
GraphicsBox[
{RGBColor[1, 0, 0], LineBox[CompressedData["
1:eJxd1TeOlEEQBeCCiJCAgIBgWSGEEEJ4D9t47xbvm4XFw3KEOkofpY7SR+AI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"]]},
AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->True,
AxesLabel->{
FormBox[
TagBox["n", HoldForm], TraditionalForm],
FormBox[
RowBox[{"BL", "(",
TagBox["n", HoldForm], ")"}], TraditionalForm]},
AxesOrigin->{0, Automatic},
PlotRangeClipping->True], {160, -333.73835392494317`},
ImageScaled[{0.5, 0.5}], {360, 222.49223594996212`}], InsetBox[
GraphicsBox[
{RGBColor[1, 0, 0], LineBox[CompressedData["
1:eJxd1UcOK0UUBdCnP2LYAwYMGDRfCCGEUJMzFDlDkzMU4ZPhNzlDLaGXUEvo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"]]},
AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->True,
AxesLabel->{
FormBox[
TagBox["n", HoldForm], TraditionalForm],
FormBox[
RowBox[{"BL", "(",
TagBox["n", HoldForm], ")"}], TraditionalForm]},
AxesOrigin->{0, Automatic},
PlotRangeClipping->True], {480, -333.73835392494317`},
ImageScaled[{0.5, 0.5}], {360, 222.49223594996212`}], InsetBox[
GraphicsBox[
{RGBColor[1, 0, 0], LineBox[CompressedData["
1:eJxd2TkMw2YZBuBfiCGjB4YMHUyFqgghZO5w1kCBcJtyhauYQkuAQs0d7p87
3B4zeszoMeM/ZvSY0WNGjxlBJYufLJ+evHrzRoqiDHnt8y8/+8Krwv8erw7h
lfvKY3p6fkM5vwuc8fxr8BI/gXNe50n8OvwUXuHX4zfgN+KC9/0m/Gb8FvxW
/Db8dvwOvMbvxO/C78bvwe/F78NP4/JxH59TxOH95Dh8gByHD5Lj8Aw5Dh8i
x+HD5Dh8hByHDTkOHyXH4WPkOHycHIdPkOPwSXIcPkWOw6fJcage9/G9LHHE
CYfP0McRJxyepY8jTjh8lj6OOOHwOfo44oTD5+njiBMOX6CPI044fJE+jjjh
sKWPI044fIk+jjjh8GX6OOKEw1fo44gTDl+ljyNOOHyNPo444fAcfRxxwuHr
9HHECYf6cR+/wzkucY0j7nDCIw7fYB+XuMYRdzjhEYfn2cclrnHEHU54xOGb
7OMS1zjiDic84vAt9nGJaxxxhxMecXiBfVziGkfc4YRHHF5kH5e4xhF3OOER
h2+zj0tc44g7nPCIw459XOIaR9zhhEccvsM+LnGNI+5wwiMO32Ufl7jGEXc4
4RGH77GPS1zjiDuc8IjDS+zjEtc44g4nPOLwffZxiWsccYcTHnH4Afu4xDWO
uMMJjzi8zD4ucY0j7nDCIw7N4z75/5vhHBe4xBWucYMjbnGHe5zwgEc84fDD
uTOc4wKXuMI1bnDELe5wjxMe8IgnHH40d4ZzXOASV7jGDY64xR3uccIDHvGE
w4/nznCOC1ziCte4wRG3uMM9TnjAI55w+MncGc5xgUtc4Ro3OOIWd7jHCQ94
xBMOP507wzkucIkrXOMGR9ziDvc44QGPeMLhZ3NnOMcFLnGFa9zgiFvc4R4n
POARTzj8fO4M57jAJa5wjRsccYs73OOEBzziCYf93BnOcYFLXOEaNzjiFne4
xwkPeMQTDr+YO8M5LnCJK1zjBkfc4g73OOEBj3jC4ZdzZzjHBS5xhWvc4Ihb
3OEeJzzgEU84/GruDOe4wCWucI0bHHGLO9zjhAc84gmHX8+d4RwXuMQVrnGD
I25xh3uc8IBHPOHwm7kznOMCl7jCNW5wxC3ucI8THvCIJxx+O3eGc1zgEle4
xg2OuMUd7nHCAx7xhMPv5s5wjgtc4grXuMERt7jDPU54wCOecIiP+/iffYEz
vMQ5XuECr3GJN7jCW1zjHW7wHkd8wC0+4g6fcI/POOELHvAVj/iGJ3zH4fdz
L3CGlzjHK1zgNS7xBld4i2u8ww3e44gPuMVH3OET7vEZJ3zBA77iEd/whO84
/GHuBc7wEud4hQu8xiXe4ApvcY13uMF7HPEBt/iIO3zCPT7jhC94wFc84hue
8B2HP869wBle4hyvcIHXuMQbXOEtrvEON3iPIz7gFh9xh0+4x2ec8AUP+IpH
fMMTvuPwp7kXOMNLnOMVLvAal3iDK7zFNd7hBu9xxAfc4iPu8An3+IwTvuAB
X/GIb3jCdxz+PPcCZ3iJc7zCBV7jEm9whbe4xjvc4D2O+IBbfMQdPuEen3HC
FzzgKx7xDU/4jsNf5l7gDC9xjle4wGtc4g2u8BbXeIcbvMcRH3CLj7jDJ9zj
M074ggd8xSO+4Qnfcfjr3Auc4SXO8QoXeI1LvMEV3uIa73CD9zjiA27xEXf4
hHt8xglf8ICveMQ3POE7Doe5FzjDS5zjFS7wGpd4gyu8xTXe4QbvccQH3OIj
7vAJ9/iME77gAV/xiG94wncc/jb3Amd4iXO8wgVe4xJvcIW3uMY73OA9jviA
W3zEHT7hHp9xwhc84Cse8Q1P+I7D3+de4AwvcY5XuMBrXOINrvAW13iHG7zH
ER9wi4+4wyfc4zNO+IIHfMUjvuEJ33H4x9wLnOElzvEKF3iNS7zBFd7iGu9w
g/c44gNu8RF3+IR7fMYJX/CAr3jENzzhOw7/nHuBM7zEOV7hAq9xiTe4wltc
4x1u8B5HfMAtPuIOn3CPzzjhCx7wFY/4hid8x+Ffcy9whpc4xytc4DUu8QZX
eItrvMMN3uOID7jFR9zhE+7xGSd8wQO+4hHf8ITvOPx77gXO8BLneIULvMYl
3uAKb3GNd7jBexzxAbf4iDt8wj0+44QveMBXPOIbnvAdh//MvcAZXuIcr3CB
17jEG1zhLa7xDjd4jyM+4BYfcYdPuMdnnPAFD/iKR3zDE77j0D7uU+V/AYiy
MKA=
"]]},
AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->True,
AxesLabel->{
FormBox[
TagBox["n", HoldForm], TraditionalForm],
FormBox[
RowBox[{"BL", "(",
TagBox["n", HoldForm], ")"}], TraditionalForm]},
AxesOrigin->{0, Automatic},
PlotRangeClipping->True], {800, -333.73835392494317`},
ImageScaled[{0.5, 0.5}], {360, 222.49223594996212`}]}}, {}},
ContentSelectable->True,
ImageSize->400,
PlotRangePadding->{6, 5}]], "Output"]
}, Open ]]
}, Open ]]
}, Open ]]
},
WindowSize->{747, 892},
WindowMargins->{{187, Automatic}, {Automatic, 14}},
FrontEndVersion->"9.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (March 5, \
2013)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)
(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[579, 22, 27, 0, 92, "Title"],
Cell[CellGroupData[{
Cell[631, 26, 28, 0, 44, "Subsection"],
Cell[662, 28, 58, 3, 49, "Text"],
Cell[723, 33, 285, 9, 49, "Text"],
Cell[1011, 44, 293, 10, 30, "Text"],
Cell[1307, 56, 99, 2, 30, "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell[1443, 63, 26, 0, 44, "Subsection"],
Cell[1472, 65, 291, 8, 46, "Input"],
Cell[1766, 75, 291, 7, 46, "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell[2094, 87, 29, 0, 80, "Section"],
Cell[CellGroupData[{
Cell[2148, 91, 218, 6, 28, "Input"],
Cell[2369, 99, 63, 1, 28, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[2469, 105, 247, 7, 28, "Input"],
Cell[2719, 114, 90, 2, 28, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[2846, 121, 222, 6, 28, "Input"],
Cell[3071, 129, 169, 5, 49, "Output"]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell[3289, 140, 24, 0, 80, "Section"],
Cell[CellGroupData[{
Cell[3338, 144, 121, 3, 28, "Input"],
Cell[3462, 149, 1108, 16, 80, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[4607, 170, 123, 3, 28, "Input"],
Cell[4733, 175, 1108, 16, 80, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[5878, 196, 124, 3, 70, "Input"],
Cell[6005, 201, 1108, 16, 70, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[7150, 222, 155, 5, 70, "Input"],
Cell[7308, 229, 121, 4, 70, "Output"]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell[7478, 239, 24, 0, 70, "Section"],
Cell[CellGroupData[{
Cell[7527, 243, 1073, 30, 70, "Input"],
Cell[8603, 275, 8210, 171, 70, "Output"]
}, Open ]]
}, Open ]]
}, Open ]]
}
]
*)
(* End of internal cache information *)
