%I #19 Jan 10 2017 08:09:03
%S 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 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 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 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 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 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 Here only primitive Pythagorean triangles with even leg b and area A given in A249869 are considered. See A249866, also for references.
%H 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 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 The triangle T(n, m) begins:
%e n\m 1 2 3 4 5 6 7 8 9 10
%e 2: 1
%e 3: 0 7
%e 4: -7 0 17
%e 5: 0 -1 0 31
%e 6: -23 0 0 0 49
%e 7: 0 -17 0 23 0 71
%e 8: -47 0 -7 0 41 0 97
%e 9: 0 -41 0 7 0 0 0 127
%e 10: -79 0 -31 0 0 0 89 0 161
%e 11: 0 -73 0 -17 0 47 0 119 0 199
%e n\m 1 2 3 4 5 6 7 8 9 10
%e ...
%e n = 12: -119 0 0 0 1 0 73 0 0 0 241,
%e n = 13: 0 -113 0 -49 0 23 0 103 0 191 0 287,
%e n = 14: -167 0 -103 0 -31 0 0 0 137 0 233 0 337,
%e n = 15: 0 -161 0 -89 0 0 0 79 0 0 0 0 0 391.
%e ...
%e The triples (2*r(n, m),2*s(n, m),2*t(n, m)) begin (we abbreviate here (0,0,0) by 0):
%e n\m 1 2 3 4 ...
%e 2: (1,5,7)
%e 3: 0 (7,13,17)
%e 4: (-7,17,23) 0 (17,25,31)
%e 5: 0 (-1,29,41) 0 (31,41,49)
%e ...
%e n = 6: (-23, 37, 47) 0 0 0 (49,61,71),
%e n = 7: 0 (-17,53,73) 0 (23,65,89) 0 (71,85,97),
%e n = 8: (-47,65,79) 0 (-7,73,103) 0 (41,89,119) 0 (97,113,127),
%e n = 9: 0 (-41,85,113) 0 (7,97,137) 0 0 0 (127,145,161),
%e n =10: (-79, 101, 119) 0 (-31,109,151) 0 0 0 (89,149,191) 0 (161,181,199).
%e ...
%e 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 n = 2: (1/4,25/4,49/4;6),
%e n = 3: 0 (49/4,169/4,289/4;30),
%e n = 4: (49/4,289/4,529/4;60) 0 (289/4,625/4,961/4;84),
%e n = 5: 0 (1/4,841/4,1681/4;210) 0 (961/4,1681/4,2401/4;180),
%e n = 6: (529/4,1369/4,2209/4;210) 0 0 0 (2401/4,3721/4,5041/4;330),
%e 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 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 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 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 ...
%Y Cf. A222946, A225949, A249866, A249869.
%K sign,tabl,easy
%O 2,3
%A _Wolfdieter Lang_, Nov 29 2016