I knew nothing about generating Pythagorean triples in 2009 so I looked for them in a spreadsheet. Millions of formulas later, I found a pattern of sets shown in the sample below.
$$\begin{array}{c|c|c|c|c|}
Set_n & Triple_1 & Triple_2 & Triple_3 & Triple_4 \\ \hline
Set_1 & 3,4,5 & 5,12,13& 7,24,25& 9,40,41\\ \hline
Set_2 & 15,8,17 & 21,20,29 &27,36,45 &33,56,65\\ \hline
Set_3 & 35,12,37 & 45,28,53 &55,48,73 &65,72,97 \\ \hline
Set_{4} &63,16,65 &77,36,85 &91,60,109 &105,88,137\\ \hline
\end{array}$$
In each $Set_n$, $(C-B)=(2n-1)^2$, the increment between consecutive values of $A$ is $2(2n-1)k$ where $k$ is the member number or count within the set, and $A=(2n-1)^2+2(2n-1)k$. I solved the Pythagorean theorem for $B$ and $C$, substituted now-known the expressions for $A$ and $(C-B)$, and got $\quad B=2(2n-1)k+2k^2\qquad C=(2n-1)^2+2(2n-1)k+2k^2$.
I have since learned the my formula is the equivalent of replacing $(m,n)$ in Euclid's formula with $((2n-1+k),k)$. I found ways of using either my formula or Euclid's to find triples given only sides, perimeters, ratios, and areas as well as polygons and pyramids constructed of dissimilar primitive triples.
I found that the first member of each set $(k=1)$ and all members of $Set_1 (n=1)$ are primitive. I found that, if $(2n-1)$ is prime, only primitives will be generated in $Set_n$ if $A=(2n-1)^2+2(2n-1)k+\bigl\lfloor\frac{k-1}{2n-2}\bigr\rfloor $ and I found that, if $(2n-1)$ is composite, I could obtain only primitives in $Set_n$ by generating and subtracting the set of [multiple] triples generated when $k$ is a $1$-or-more multiple of any factor of $(2n-1)$. The primitive count in the former is obtained directly; the count for the latter is obtained by combinatorics.
I'm trying to write a paper "On Finding Pythagorean Triples". Surely someone has discovered these sets in the $2300$ years since Euclid but I haven't found and reference to them or any subsets of Pythagorean triples online or in the books I've bought and read. So my question is: "Where have these distinct sets of triples been mentioned before?" I would like to cite the work if I can find it.
The bounty just expired and neither of the two answers has been helpful. I have not quite a day to award the bounty. Any takers? Where and when have these sets been discovered before?
Best Answer
In L. E. Dickson, History of the Theory of Numbers, Volume II, page 167
The footnote 18 is briefly "Hist. Acad. Sc. Paris, 1729, 318."
Your formulas are
$$A\!=\!(2n\!-\!1)^2\!+\!2(2n\!-\!1)k,\\ B\!=\!2(2n\!-\!1)k\!+\!2k^2,\\ C\!=\!(2n\!-\!1)^2\!+\!2(2n\!-\!1)k\!+\!2k^2.$$
Get this from Lagny's formulas if $\,d\,$ is replaced by $\,2n-1\,$ and $\,n\,$ is replaced with $\,k.\,$
Thus, your formula is equivalent to de Lagny's except $\,2n-1=d\,$ is always odd, however, if $\,d\,$ is even, the triple has a common factor of $2$ and can not be primitive.