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 A213188 #21 Jan 05 2025 19:51:39
%S A213188 10,45,136,325,435,595,630,666,780,1225,2080,2145,3321,5050,5565,5886,
%T A213188 6216,7381,7503,9316,10440,11026,11175,12246,13530,14196,14365,14535,
%U A213188 15753,16653,18915,19306,24310,25425,32896,33670,39060,41905,42195,49141,50721,52650
%N A213188 Triangular numbers that are hypotenuse and a leg of a Pythagorean triple.
%C A213188 The square of the third leg is a sum of consecutive cubes (or one cube). See A126200, A217843. In the Pythagorean triple {325,91,312}, 312^2 = 14^3 + 15^3 + ... + 25^3 = 97344.
%C A213188 It is possible for both of the legs to be triangular numbers as well as the hypotenuse. The only known example is 8778^2 + 10296^2 = 13530^2. - _Andrew Howroyd_, Aug 17 2018
%H A213188 D. W. Ballew, R. C. Weger, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/17-2/ballew.pdf">Pythagorean Triples and Triangular Numbers</a>, Fibonacci Quarterly, 17.2 (1979), 168-171.
%e A213188 The triangular numbers 45 and 36 are the hypotenuse and leg of a Pythagorean triple {45, 36, 27}.
%o A213188 (PARI) {for(i=1,10^3,k=1;v=1;a=i*(i+1)/2;while(k<=i-1&&v,b=k*(k+1)/2;if(issquare(a*a-b*b),v=0;print1(a,", "));k+=1))}
%Y A213188 Cf. A126200, A213189, A217843.
%K A213188 nonn
%O A213188 1,1
%A A213188 _Antonio Roldán_, Feb 28 2013