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 A291591 #36 Aug 08 2021 01:44:22 %S A291591 71831760,73513440,1675212000,6913932480,4323749790360,2678930100000, %T A291591 175434192299520,503151375767040 %N A291591 Numbers k such that there exist exactly five distinct Pythagorean triangles, at least one of them primitive, with area k. %C A291591 I solve x^2 + 3*y^2 = (2*r)^2 over the positive integers. q, r, q-p and p are the y-coordinates in the first quadrant. Area = q*r*(q-p)*p. There are three Pythagorean triangles with this area. j, x, y with x > y and Area = j^2*x*y*(x-y)*(x+y) gives the area of an Pythagorean triangle. %C A291591 Example: r = 169 in x^2 + 3*y^2 = (2*169)^2 gives q = 176, r = 169, q-p = 161 and p = 15; %C A291591 k = q*r*(q-p)*p = 176*169*161*15 = 71831760. %C A291591 j = 26, x = 23, y = 12 and j = 26, x = 28, y = 5 gives two Pythagorean triangles with k = 71831760; %C A291591 k = 676*23*12*11*35 = 71831760 and k = 676*28*5*23*33 = 71831760. %e A291591 p^2 - p*q + q^2 = r^2; %e A291591 p = 115, q = 448, q-p = 333, r = 403; %e A291591 k = p*q*(q-p)*r = 115*448*333*403 = 6913932480. %e A291591 x = 414, y = 104 and x = 558, y = 40 gives the same area. %e A291591 k = x*y*(x-y)*(x+y) = 414*104*310*518 = 6913932480. %e A291591 k = x*y*(x-y)*(x+y) = 558*40*518*598 = 6913932480. %Y A291591 Cf. A055193. %K A291591 nonn,more %O A291591 1,1 %A A291591 _Sture Sjöstedt_, Aug 27 2017 %E A291591 a(2), a(4)-a(7) from _Giovanni Resta_, Aug 28 2017 %E A291591 Missing term 73513440 inserted by _Miguel-Ángel Pérez García-Ortega_, Jul 19 2021