Definition:Almost Isosceles Pythagorean Triangle

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Definition

Let $P$ be a Pythagorean triangle whose legs are $a$ and $b$.

Then $P$ is almost isosceles if and only if $\size {a - b} = 1$


That is, if the legs of $P$ differ by $1$.


Sequence

The sequence of almost isosceles Pythagorean triangles can be tabulated as follows:

$\begin{array} {r r | r r | r r r} m & n & m^2 & n^2 & 2 m n & m^2 - n^2 & m^2 + n^2 \\ \hline 2 & 1 & 4 & 1 & 4 & 3 & 5 \\ 5 & 2 & 25 & 4 & 20 & 21 & 29 \\ 12 & 5 & 144 & 25 & 120 & 119 & 169 \\ 29 & 12 & 841 & 144 & 696 & 697 & 985 \\ 70 & 29 & 4900 & 841 & 4060 & 4059 & 5741 \\ 169 & 70 & 28 \, 561 & 4900 & 23 \, 660 & 23 \, 661 & 33 \, 461 \\ \hline \end{array}$

The sequence of elements of the generators are the Pell numbers:

$1, 2, 5, 12, 29, 70, 169, 408, \ldots$


Sources