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 A278717 #19 Jan 10 2017 08:09:03 %S A278717 1,0,7,-7,0,17,0,-1,0,31,-23,0,0,0,49,0,-17,0,23,0,71,-47,0,-7,0,41,0, %T A278717 97,0,-41,0,7,0,0,0,127,-79,0,-31,0,0,0,89,0,161,0,-73,0,-17,0,47,0, %U A278717 119,0,199,-119,0,0,0,1,0,73,0,0,0,241,0,-113,0,-49,0,23,0,103,0,191,0,287,-167,0 %N A278717 Triangle read by rows: T(n, m) gives the difference between the even and odd leg of the primitive Pythagorean triangle determined by (n, m) with n > m >= 1, gcd(n, m) = 1 and n+m odd, or 0 for other (n, m). %C A278717 T(n, m) also gives twice the member r(n, m) of the triple (r(n, m), s(n, m), t(n, m)) with squares r(n, m)^2, s(n, m)^2 and t(n, m)^2 in arithmetic progression with common difference A(n, m) = A249869(n, m), the area of the primitive Pythagorean triangle. The members 2*s(n, m) (hypotenuse) and 2*t(n, m) (sum of catheti) are given in A222946(n, m) and A225949(n, m), respectively. %C A278717 There is a one-to-one correspondence between rational Pythagorean triangles (a,b,c) with area A and three squares r^2, s^2 and t^2 in arithmetic progression with common difference A > 0: (r, s, t) = ((b-a)/2, c/2, (b+a)/2) and (a,b,c) = (t-r, t+r, 2*s). See the Keith Conrad link, Theorem 3.1. Leg exchange leads to the same progression of squares but r is exchanged with -r. %C A278717 Here only primitive Pythagorean triangles with even leg b and area A given in A249869 are considered. See A249866, also for references. %H A278717 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 A278717 T(n, m) = 2*n*m - (n^2 - m^2) if n > m >= 1 and gcd(n, m) = 1, n+m odd, and T(n, m) = 0 otherwise. %e A278717 The triangle T(n, m) begins: %e A278717 n\m 1 2 3 4 5 6 7 8 9 10 %e A278717 2: 1 %e A278717 3: 0 7 %e A278717 4: -7 0 17 %e A278717 5: 0 -1 0 31 %e A278717 6: -23 0 0 0 49 %e A278717 7: 0 -17 0 23 0 71 %e A278717 8: -47 0 -7 0 41 0 97 %e A278717 9: 0 -41 0 7 0 0 0 127 %e A278717 10: -79 0 -31 0 0 0 89 0 161 %e A278717 11: 0 -73 0 -17 0 47 0 119 0 199 %e A278717 n\m 1 2 3 4 5 6 7 8 9 10 %e A278717 ... %e A278717 n = 12: -119 0 0 0 1 0 73 0 0 0 241, %e A278717 n = 13: 0 -113 0 -49 0 23 0 103 0 191 0 287, %e A278717 n = 14: -167 0 -103 0 -31 0 0 0 137 0 233 0 337, %e A278717 n = 15: 0 -161 0 -89 0 0 0 79 0 0 0 0 0 391. %e A278717 ... %e A278717 The triples (2*r(n, m),2*s(n, m),2*t(n, m)) begin (we abbreviate here (0,0,0) by 0): %e A278717 n\m 1 2 3 4 ... %e A278717 2: (1,5,7) %e A278717 3: 0 (7,13,17) %e A278717 4: (-7,17,23) 0 (17,25,31) %e A278717 5: 0 (-1,29,41) 0 (31,41,49) %e A278717 ... %e A278717 n = 6: (-23, 37, 47) 0 0 0 (49,61,71), %e A278717 n = 7: 0 (-17,53,73) 0 (23,65,89) 0 (71,85,97), %e A278717 n = 8: (-47,65,79) 0 (-7,73,103) 0 (41,89,119) 0 (97,113,127), %e A278717 n = 9: 0 (-41,85,113) 0 (7,97,137) 0 0 0 (127,145,161), %e A278717 n =10: (-79, 101, 119) 0 (-31,109,151) 0 0 0 (89,149,191) 0 (161,181,199). %e A278717 ... %e A278717 The quartets (r(n,m)^2,s(n,m)^2,t(n,m)^2;A(n, m)) of squares in arithmetic progression with common difference A(n,m) = A249869(n,m) begin (here (0,0,0;0) is abbreviated as 0): %e A278717 n = 2: (1/4,25/4,49/4;6), %e A278717 n = 3: 0 (49/4,169/4,289/4;30), %e A278717 n = 4: (49/4,289/4,529/4;60) 0 (289/4,625/4,961/4;84), %e A278717 n = 5: 0 (1/4,841/4,1681/4;210) 0 (961/4,1681/4,2401/4;180), %e A278717 n = 6: (529/4,1369/4,2209/4;210) 0 0 0 (2401/4,3721/4,5041/4;330), %e A278717 n = 7: 0 (289/4,2809/4,5329/4;630) 0 (529/4,4225/4,7921/4;924) 0 (5041/4,7225/4,9409/4;546), %e A278717 n = 8: (2209/4,4225/4,6241/4;504) 0 (49/4,5329/4,10609/4;1320) 0 (1681/4,7921/4,14161/4;1560) 0, (9409/4,12769/4,16129/4;840), %e A278717 n = 9: 0, (1681/4,7225/4,12769/4;1386) 0 (49/4,9409/4,18769/4;2340) 0 0 0 (16129/4,21025/4,25921/4;1224) %e A278717 n = 10: (6241/4,10201/4,14161/4);990) 0 (961/4,11881/4,22801/4;2730) 0 0 0 (7921/4,22201/4,36481/4;3570) 0 (25921/4,32761/4,39601/4;1710). %e A278717 ... %Y A278717 Cf. A222946, A225949, A249866, A249869. %K A278717 sign,tabl,easy %O A278717 2,3 %A A278717 _Wolfdieter Lang_, Nov 29 2016