CoCalc
Sharedlatex-tests / example.texOpen in CoCalc
Authors: harald @schil.ly, Testing CoCalc, John Jeng, Harald Schilly, ℏal Snyder, William A. Stein
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% General example LaTeX file for including Sage calculations and plots
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% Build with:
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%
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% (pdf)latex example.tex; sage example.sage; pdflatex example.tex
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%
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% Please read README and the documentation of the SageTeX package for
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% more information!
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\documentclass{article}
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\title{Examples of embedding Sage in \LaTeX{} with \textsf{Sage\TeX}}
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\author{Drake and others}
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\usepackage{amsmath}
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\usepackage{sagetex}
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%
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% If you want SageTeX to use Imagemagick's `convert' utility to make eps
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% files from png files when generating a dvi file, add the "imagemagick"
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% option above:
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%
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% \usepackage[imagemagick]{sagetex}
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\setlength{\sagetexindent}{10ex}
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\begin{document}
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\maketitle
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\section{Inline SageMath, code blocks}
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This is an example $2+2=\sage{2+2}$. If you raise the current year mod
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$100$ (which equals $\sage{mod(\the\year, 100)}$) to the power of the
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current day ($\the\day$), you get
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$\sage{Integer(mod(\the\year, 100))^\the\day}$.
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Also, $\the\year$ modulo $42$ is $\sage{\the\year\percent 42}$.
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Code block, which uses a variable \texttt{s} to store the solutions:
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\begin{sageblock}
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1+1
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var('a,b,c,d')
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eqn = [a+b*c==1, b-a*c==0, a+b==5]
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s = solve(eqn, a,b,c)
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\end{sageblock}
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Solutions of $\mbox{eqn}=\sage{eqn}$:
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\[
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\sage{s[0]}
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\]
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\[
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\sage{s[1]}
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\]
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Now we evaluate the following block:
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\begin{sageblock}
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E = EllipticCurve("37a")
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\end{sageblock}
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You can't do assignment inside \verb|\sage| macros, since Sage doesn't
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know how to typeset the output of such a thing. So you have to use a
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code block. The elliptic curve $E$ given by $\sage{E}$ has discriminant
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$\sage{E.discriminant()}$.
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You can do anything in a code block that you can do in Sage and/or
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Python. Here we save an elliptic curve into a file.
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\begin{sageblock}
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try:
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E = load('E2')
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except IOError:
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E = EllipticCurve([1,2,7,4,5])
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E.anlist(100000)
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E.save('E2')
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\end{sageblock}
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The 9999th Fourier coefficient of $\sage{E}$ is $\sage{E.anlist(100000)[9999]}$.
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The following code block doesn't appear in the typeset file\dots
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\begin{sagesilent}
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e = 2
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e = 3*e + 1
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\end{sagesilent}
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but we can refer to whatever we did in that code block: $e=\sage{e}$.
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\begin{sageblock}
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var('x')
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f(x) = log(sin(x)/x)
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\end{sageblock}
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The Taylor Series of $f$ begins: $\sage{ f.taylor(x, 0, 10) }$.
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\section{Plotting}
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Here's a plot of the elliptic curve $E$.
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\sageplot[scale=.4]{E.plot(-3,3)}
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\begin{sagesilent}
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# the var line is unecessary unless you've defined x to be something
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# other than a symbolic variable
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var('x')
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f(x) = -x^3+4*x^2+7*x-4
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\end{sagesilent}
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You can use variables to hold plot objects and do stuff with them.
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\begin{sageblock}
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p = plot(f, x, -5, 8)
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\end{sageblock}
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Here's a small plot of $f$ from $-5$ to $5$, which I've centered:
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\begin{center} \sageplot[scale=.2]{p} \end{center}
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On second thought, use the default size of $3/4$ the \verb|\textwidth|
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and don't use axes:
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\sageplot[scale=.4]{p, axes=False}
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Remember, you're using Sage, and can therefore call upon any of the
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software packages Sage is built out of.
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\begin{sageblock}
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f = maxima('sin(.4 * x)^3*cos(x)')
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g = f.integrate('x')
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\end{sageblock}
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Plot $g(x)$, but don't typeset it.
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\begin{sagesilent}
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# g is a Maxima thingy, it needs to get converted into a Sage object
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plot1 = plot(g.sage(),x,-1,2*pi)
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\end{sagesilent}
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You can specify a file format and options for \verb|includegraphics|.
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The default is for EPS and PDF files, which are the best choice in
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almost all situations. (Although see the section on 3D plotting.)
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\sageplot[angle=45, width=.5\textwidth][png]{plot1}
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If you use regular \verb|latex| to make a DVI file, you'll see a box,
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because DVI files can't include PNG files. If you use \verb|pdflatex|
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that will work. See the documentation for details.
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When using \verb|\sageplot|, you can pass in just about anything that
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Sage can call \verb|.save()| on to produce a graphics file:
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\begin{center}
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\sageplot{plot1 + plot(f.sage(),x,-1,2*pi,rgbcolor=hue(0.4)), figsize=[1,2]}
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\end{center}
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\sageplot{graphs.FlowerSnark().plot()}
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\begin{sageblock}
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G4 = DiGraph({1:[2,2,3,5], 2:[3,4], 3:[4], 4:[5,7], 5:[6]},\
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multiedges=True)
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G4plot = G4.plot(layout='circular')
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\end{sageblock}
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\sageplot{G4plot, axes=False}
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Indentation and so on works fine.
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\begin{sageblock}
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s = 7
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s2 = 2^s
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P.<x> = GF(2)[]
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M = matrix(parent(x),s2)
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for i in range(s2):
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p = (1+x)^i
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pc = p.coeffs()
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a = pc.count(1)
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for j in range(a):
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idx = pc.index(1)
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M[i,idx+j] = pc.pop(idx)
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matrixprogram = matrix_plot(M,cmap='Greys')
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\end{sageblock}
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And here's the picture:
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\sageplot{matrixprogram}
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Reset \texttt{x} in Sage so that it's not a generator for the polynomial
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ring: \sage{var('x')}
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\subsection{3D plotting}
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3D plotting right now is problematic because there's no convenient way
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to produce vector graphics. We can make PNGs, though, and since the
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\verb|sageplot| command defaults to EPS and PDF, \emph{you must specify
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a valid format for 3D plotting}. Sage right now (version 3.4.2) can't
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produce EPS or PDF files from plot3d objects, so if you don't specify a
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valid format, things will go badly. You can specify the
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``\texttt{imagemagick}'' option, which will use the Imagemagick
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\texttt{convert} utility to make EPS files. See the documentation for
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details.
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Here's the famous Sage cube graph:
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\begin{sageblock}
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G = graphs.CubeGraph(5)
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\end{sageblock}
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% need empty [] so sageplot knows you want png format, and aren't
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% passing an option to includegraphics
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\sageplot[][png]{G.plot3d(engine='tachyon')}
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\section{Pausing Sage\TeX}
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\label{sec:pausing-sagetex}
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Sometimes you want to ``pause'' for a bit while writing your document if
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you have embedded a long calculation or just want to concentrate on the
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\LaTeX{} and ignore any Sage stuff. You can use the \verb|\sagetexpause|
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and \verb|\sagetexunpause| macros to do that.
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\sagetexpause
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A calculation: $\sage{factor(2^325 + 1)}$ and a code environment that
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simulates a time-consuming calculation. While paused, this will get
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skipped over.
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\begin{sageblock}
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import time
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time.sleep(15)
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\end{sageblock}
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Graphics are also skipped: \sageplot{plot(2*sin(x^2) + x^2, (x, 0, 5))}
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\sagetexunpause
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\section{Make Sage write your \LaTeX{} for you}
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With \textsf{Sage\TeX}, you can not only have Sage do your math for you,
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it can write parts of your \LaTeX{} document for you! For example, I
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hate writing \texttt{tabular} environments; there's too many fiddly
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little bits of punctuation and whatnot\ldots and what if you want to add
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a column? It's a pain---or rather, it \emph{was} a pain. Here's how to
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make Pascal's triangle. It requires the \texttt{amsmath} package because
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of what Sage does when producing a \LaTeX{} representation of a string.
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(It puts it inside a \verb|\text| macro.)
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%\begin{sageblock}
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%def pascals_triangle(n):
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% # start of the table
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% s = r"\begin{tabular}{cc|" + "r" * (n+1) + "}"
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% s += r" & & $k$: & \\"
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% # second row, with k values:
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% s += r" & "
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% for k in [0..n]:
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% s += "& %d " % k
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% s += r"\\"
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% # the n = 0 row:
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% s += r"\hline" + "\n" + r"$n$: & 0 & 1 & \\"
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% # now the rest of the rows
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% for r in [1..n]:
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% s += " & %d " % r
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% for k in [0..r]:
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% s += "& %d " % binomial(r, k)
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% s += r"\\"
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% # add the last line and return
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% s += r"\end{tabular}"
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% return s
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%
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% # how big should the table be?
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% n = 8
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%\end{sageblock}
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Okay, now here's the table. To change the size, edit \texttt{n} above.
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If you have several tables, you can use this to get them all the same
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size, while changing only one thing.
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%\begin{center}
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%$\sage{pascals_triangle(n)}$
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%\end{center}
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$\sage{version()}$
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\end{document}
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