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A330138 List of pairs [j,k] ordered by increasing j and k according to the procedure described in the Comments.

%I A330138 #45 Feb 25 2020 10:29:00
%S A330138 3,4,5,15,8,17,5,12,13,35,12,37,21,20,29,7,24,25,63,16,65,45,28,53,27,
%T A330138 36,45,9,40,41,99,20,101,77,36,85,55,48,73,33,56,65,11,60,61,143,24,
%U A330138 145,117,44,125,91,60,109,65,72,97,39,80,89,13,84,85
%N A330138 List of pairs [j,k] ordered by increasing j and k according to the procedure described in the Comments.
%C A330138 Procedure: The sequence lists two indices, j and k. Let (a, b, and c) represent the two legs and the hypotenuse of a right-angled triangle. The required computations are as follows:
%C A330138 (1) compute  2*j^2    (call this p, it is c-a)
%C A330138 (2) compute (2*k-1)^2 (call this q, it is c-b)
%C A330138 (3) compute 2j*(2k-1) (call this r, it is a+b-c)
%C A330138 So c = p+q+r; a = c-p; b = c-q.
%C A330138 Most, but not all, of the Pythagorean triples generated here are primitive (i.e., they have sides that are relatively prime). In the first 105 Pythagorean triples, there are 14 that are not primitive. The first exception is the 9th in the list: 27,36,45, where j=3 and k=2.
%H A330138 Wikipedia, <a href="https://en.wikipedia.org/wiki/Formulas_for_generating_Pythagorean_triples#Dickson&#39;s_method">Dickson's method of generating Pythagorean triples.</a>
%e A330138 As the values of j and k increase, so do the lengths of the sides of the triangles. So it makes sense to order the triangles based on the sum (j+k).  When j=k=1, the sum (j+k)=2, and there is only one triangle (3,4,5).  When the sum of j and k is 3, there are two possibilities: j=1, k=2 (15,8,17) or j=2, k=1 (5,12,13).  When the sum is 4, there are three possibilities. Each time the sum (j+k) increases by one, the number of triangles also increases by one. And the total number of triangles grows according to sequence A000217 (the triangular numbers).
%e A330138 Value of j+k:    2, 3, 4,  5,  6,  7,  8,  9, 10, 11, 12, ...
%e A330138 Total triangles: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...
%e A330138 So it is clear that triangle number 61 would be in the range of triangles ranked 56 through 66, where (j+k)=12.  In fact, exactly 6 beyond the triangle associated with the values j=10, k=1.  So j=6, k=6 is associated with the sixty-first triangle. And the computations give us this Pythagorean triple: a=253, b=204, c=325.
%t A330138 Table[Map[{#2 + #3, #1 + #3, #1 + #2 + #3} & @@ {2 #1^2, (2 #2 - 1)^2, 2 #1 (2 #2 - 1)} & @@ {#, n - #} &, Range[n - 1]], {n, 7}] // Flatten (* _Michael De Vlieger_, Dec 14 2019 *)
%Y A330138 Cf. A103606.
%K A330138 nonn,tabf
%O A330138 1,1
%A A330138 _Joseph Damico_, Dec 02 2019