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 A278711 #26 Feb 22 2025 12:07:41 %S A278711 12,0,45,240,0,112,0,525,0,225,1260,0,0,0,396,0,2205,0,1617,0,637, %T A278711 4032,0,3520,0,2496,0,960,0,6237,0,5265,0,0,0,1377,9900,0,9100,0,0,0, %U A278711 5100,0,1900,0,14157,0,12705,0,10285,0,6897,0,2541,20592,0,0,0,17136,0,13680,0,0,0,3312,0,27885,0,25857,0,22477,0,17745,0,11661,0,4225,38220,0,36652,0,33516,0,0,0,22540,0,14700,0,5292,0,49725,0,47025,0,0,0,36225,0,0,0,0,0,6525 %N A278711 Triangle T read by rows: T(n, m), for n >= 2, and m=1, 2, ..., n-1, equals the positive integer solution x of y^2 = x^3 - A(n, m)^2*x with the area A(n, m) = A249869(n, m) of the primitive Pythagorean triangle characterized by (n, m) or 0 if no such triangle exists. %C A278711 The corresponding triangle with the square root of the positive integer solutions y is A278712. %C A278711 A primitive Pythagorean triangle is characterized by two integers n > m >= 1, gcd(n, m) = 1 and n+m odd. See A249866, also for references. %C A278711 For the one-to-one correspondence between rational Pythagorean triangles with area A > 0 and rational points on the elliptic curve y^2 = x^3 - A^2*x with y not vanishing see Theorem 4.1 of the Keith Conrad link or Theorem 15.6, p. 212, of the Ash-Gross reference. %H A278711 Avner Ash and Robert Gross, <a href="https://www.jstor.org/stable/j.ctt7sx3k">Elliptic tales: curves, counting, and number theory</a>, Princeton University Press, 2012. %H A278711 Keith Conrad, <a href="http://www.math.uconn.edu/~kconrad/articles/congruentnumber.pdf">The Congruent Number Problem</a>, The Harvard College Mathematics Review, 2008. %F A278711 T(n, m) = (n^2 - m^2)*n^2 if n > m >= 1, gcd(n, m) = 1 and n+m is odd, and T(n, m) = 0 otherwise. %e A278711 The triangle T(n, m) begins: %e A278711 n\m 1 2 3 4 5 6 7 8 %e A278711 2: 12 %e A278711 3: 0 45 %e A278711 4: 240 0 112 %e A278711 5: 0 525 0 225 %e A278711 6: 1260 0 0 0 396 %e A278711 7: 0 2205 0 1617 0 637 %e A278711 8: 4032 0 3520 0 2496 0 960 %e A278711 9 0 6237 0 5265 0 0 0 1377 %e A278711 ........................................... %e A278711 n = 10: 9900 0 9100 0 0 0 5100 0 1900, %e A278711 n = 11: 0 14157 0 12705 0 10285 0 6897 0 2541, %e A278711 n = 12: 20592 0 0 0 17136 0 13680 0 0 0 3312, %e A278711 n = 13: 0 27885 0 25857 0 22477 0 17745 0 11661 0 4225, %e A278711 n = 14: 38220 0 36652 0 33516 0 0 0 22540 0 14700 0 5292, %e A278711 n = 15: 0 49725 0 47025 0 0 0 36225 0 0 0 0 0 6525. %e A278711 ... %e A278711 The triangle of solutions [x,y] begins ([0,0] if there is no primitive Pythagorean): %e A278711 n\m 1 2 3 4 %e A278711 2: [12,36] %e A278711 3: [0,0] [45,225] %e A278711 4:[240,3600] [0,0] [112,784] %e A278711 5: [0,0] [525,11025] [0,0] [225, 2025] %e A278711 ... %e A278711 n=6: [1260,44100] [0,0] [0,0] [0,0] [396,4356], %e A278711 n=7: [0,0] [2205,99225] [0,0] [1617,53361] [0.0] [637,8281], %e A278711 n=8: [4032,254016] [0,0] [3520,193600] [0,0] [2496,97344] [0,0] [960,14400], %e A278711 n=9: [0,0] [6237,480249] [0,0] [5265,342225] [0,0] [0,0] [0,0] [1377,23409], %e A278711 n=10: [9900,980100] [0,0] [9100,828100] [0,0] [0,0] [0,0] [5100,260100] [0,0] [1900, 36100]. %e A278711 ... %Y A278711 Cf. A249866, A249869, A278712. %K A278711 nonn,tabl,easy %O A278711 2,1 %A A278711 _Wolfdieter Lang_, Nov 27 2016