test2

genfrac{left-delim}{right-delim}{thickness}{mathstyle}
{numerator}{denominator}

    \[\HVline{8}{1}{9}{0}{10}{-9} \]

    \[\Hline{a}{b}{c}{d}\]

    \[\quadmatrix{+A}{+B}{+B}{+A}\]

\tHVline{8}{1}{9}{0}{10}{-9}

\Hline{a}{b}{c}{d}
\quadmatrix{+A}{+B}{+B}{+A}

Like this: \genfrac{}{}{0.3}{0}{a \quad b}{c \quad d}\Big| \genfrac{}{}{0}{0}{e}{}

    \[ \frac{a\quad b}{c\quad d}\Big|\frac{e}{f} \]

[a+AB b-A^2]
[c-A^2 d+AB]
a = \Big[\begin{matrix}1 & 1\\ 0 & 1\end{matrix}\Big]
Moving north-east A times produces: \Big[\begin{matrix}a+AB & b-A^2\\ c-A^2 & d+AB\end{matrix}\Big]

a = \big[ \begin{smallmatrix}1 & 1\\ 0 & 1\end{smallmatrix}\big].
Moving north-east A times produces: \big[ \begin{smallmatrix}a+AB & b-A^2\\ c-A^2 & d+AB\end{smallmatrix}\big].
How to construct:
Start with any h-line: four squares along a horizontal line with a+b=c+d, e.g. 3+4=2+5:

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Extend the arithmetic progressions along the diagonals:

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Then fill in a diagonal next to one of these 2, with any arithmetic progression. e.g:

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Then all the empty squares have been determined and can be filled in, making the sums a+b=c+d equal across all line segments (the h-lines and v-lines).
Then when drawing the squaring, draw the squares of negative size on the other side of the line from where they ‘should’ be, e.g. 2+5=10-3:

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It seems that any grid of numbers constructed this way makes a tesselation of the plane.

Also, it seems that all the 0-sized squares (I mark them with a small circle on the squaring) lie in a straight line, no matter what sequences are used. I’m sure there’s an obvious reason for that!

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Unfortunately TikZ has too many bugs with barycentric plotting. ARGGHH! Unusable. Not slight – massive and weird errors.

    \[\cfrac{\dots *}{*+\cfrac{*}{*+\cfrac{a}{b+\cfrac{c}{*+\cfrac{*}{*+\dots}}}}}=\cfrac{\dots *}{*+\cfrac{*}{*+\cfrac{am}{bm+\cfrac{cm}{*+\cfrac{*}{*+\dots}}}}}\]

Nonregular CFs can also be represented as squares in a rectangle.

    \[e=1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+ \cfrac{1}{14+\cfrac{1}{18+\dots}}}}}\]

    \[C_3=1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10}}}=\frac{193}{71}=2.7183098\dots\]

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e.g. \sqrt{3}=1+\cfrac{2}{2+\cfrac{2}{2+\cfrac{2}{2+\dots}}}
=1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\dots}}}
Picture of \phi: 1/(1+1/(1+1/(1+etc

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    \begin{align*} &x^4+10x^3+35^2+50x+24 \\ &=(x+1)(x+2)(x+3)(x+4) \\ &=\fp{(x+4)}{4} \end{align*}


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    \begin{gather*} 3\cdot 1\cdot 2+3\cdot 2\cdot 3+3\cdot 3\cdot 4+3\cdot 4\cdot 5+\dots+3\cdot 13\cdot 14=13\cdot 14\cdot 15 \\ \sum n(n+1)=\frac{n(n+1)(n+2)}{3} \end{gather*}

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