A278712 Triangle T read by rows: T(n, m), for n >= 2, and m = 1, 2, ..., n-1, equals the square root of the positive integer solution y 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.
6, 0, 15, 60, 0, 28, 0, 105, 0, 45, 210, 0, 0, 0, 66, 0, 315, 0, 231, 0, 91, 504, 0, 440, 0, 312, 0, 120, 0, 693, 0, 585, 0, 0, 0, 153, 990, 0, 910, 0, 0, 0, 510, 0, 190, 0, 1287, 0, 1155, 0, 935, 0, 627, 0, 231, 1716, 0, 0, 0, 1428, 0, 1140, 0, 0, 0, 276, 0, 2145, 0, 1989, 0, 1729, 0, 1365, 0, 897, 0, 325, 2730, 0, 2618, 0, 2394, 0, 0, 0, 1610, 0, 1050, 0, 378, 0, 3315, 0, 3135, 0, 0, 0, 2415, 0, 0, 0, 0, 0, 435
Offset: 2
Examples
The triangle T(n, m) begins: n\m 1 2 3 4 5 6 7 8 9 10 2: 6 3: 0 15 4: 60 0 28 5: 0 105 0 45 6: 210 0 0 0 66 7: 0 315 0 231 0 91 8: 504 0 440 0 312 0 120 9: 0 693 0 585 0 0 0 153 10: 990 0 910 0 0 0 510 0 190 11: 0 1287 0 1155 0 935 0 627 0 231 ... n = 12: 1716 0 0 0 1428 0 1140 0 0 0 276, n = 13: 0 2145 0 1989 0 1729 0 1365 0 897 0 325, n = 14: 2730 0 2618 0 2394 0 0 0 1610 0 1050 0 378, n = 15: 0 3315 0 3135 0 0 0 2415 0 0 0 0 0 435. ... For the solutions [x,y] see A278711.
Crossrefs
Cf. A278711.
Formula
T(n, m) = (n^2 - m^2)*n if n > m >= 1, gcd(n, m) = 1 and n+m is odd, and T(n, m) = 0 otherwise.
Comments