This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A208855 #25 Feb 12 2025 07:05:21
%S A208855 4,12,8,20,24,12,28,40,0,16,36,56,60,48,20,44,72,84,80,60,24,52,88,0,
%T A208855 112,0,0,28,60,104,132,144,140,120,84,32,68,120,156,176,180,168,140,
%U A208855 96,36,76,136,0,208,220,0,0,160,0,40
%N A208855 Array of even catheti of primitive Pythagorean triangles when read by SW-NE diagonals.
%C A208855 See the comments, reference and links in A208853. The present array is b(n,m) = 2*(2*n-1)*(2*m) if gcd(2*n-1,2*m)=1 and 0 otherwise. u=2*n-1, v=2*m. The array read by SW-NE diagonals is T(n,m):=b(n-m+1,m), n>=m>=1.
%C A208855 All primitive Pythagorean triples are given by
%C A208855 (a(n,m)=A208854(n,m),b(n,m),c(n,m)= A208853(n,m)), n>=1, m>=1. If the entry is 0 there is no primitive Pythagorean triple for these n and m values.
%H A208855 Paolo Xausa, <a href="/A208855/b208855.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150 of triangle, flattened).
%F A208855 T(n,m)=b(n-m+1,m), n>=m>=1, with b(n,m) = 4*(2*n-1)*m if gcd(2*n-1,2*m)=1 and 0 otherwise.
%e A208855 Array b(n,m):
%e A208855 m| 1 2 3 4 5 6 7 8 9 10 ...
%e A208855 v| 2 4 6 8 10 12 14 16 18 20 ...
%e A208855 n, u
%e A208855 1, 1 4 8 12 16 20 24 28 32 36 40 ...
%e A208855 2, 3 12 24 0 48 60 0 84 96 0 120 ...
%e A208855 3, 5 20 40 60 80 0 120 140 160 180 0 ...
%e A208855 4, 7 28 56 84 112 140 168 0 224 252 280 ...
%e A208855 5, 9 36 72 0 144 180 0 252 288 0 360 ...
%e A208855 6, 11 44 88 132 176 220 264 308 352 396 440 ...
%e A208855 7, 13 52 104 156 208 260 312 364 416 468 520 ...
%e A208855 8, 15 60 120 0 240 0 0 420 480 0 0 ...
%e A208855 9, 17 68 136 204 272 340 408 476 544 612 680 ...
%e A208855 10, 19 76 152 228 304 380 456 532 608 684 760 ...
%e A208855 ...
%e A208855 Triangle T(n,m):
%e A208855 m| 1 2 3 4 5 6 7 8 9 10 ...
%e A208855 v| 2 4 6 8 10 12 14 16 18 20 ...
%e A208855 n, u
%e A208855 1, 1 4
%e A208855 2, 3 12 8
%e A208855 3, 5 20 24 12
%e A208855 4, 7 28 40 0 16
%e A208855 5, 9 36 56 60 48 20
%e A208855 6, 11 44 72 84 80 60 24
%e A208855 7, 13 52 88 0 112 0 0 28
%e A208855 8, 15 60 104 132 144 140 120 84 32
%e A208855 9, 17 68 120 156 176 180 168 140 96 36
%e A208855 10, 19 76 136 0 208 220 0 0 160 0 40
%e A208855 ...
%e A208855 For some triples see the example section of A208853.
%t A208855 A208855[n_, m_] := If[CoprimeQ[#, 2*m], 4*m*#, 0] & [2*(n-m) + 1];
%t A208855 Table[A208855[n, m], {n, 15}, {m, n}] (* _Paolo Xausa_, Feb 12 2025 *)
%Y A208855 Cf. A208853, A208854.
%K A208855 nonn,easy,tabl
%O A208855 1,1
%A A208855 _Wolfdieter Lang_, Mar 05 2012