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A278712
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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.
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1
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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
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OFFSET
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2,1
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COMMENTS
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The corresponding solutions x are given in A278711, where also details are found.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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