OFFSET
2,1
COMMENTS
See the Hardy-Wright (Theorem 225, p. 190) and Niven-Zuckerman-Montgomery (Theorem 5.5, p. 232) references for primitive Pythagorean triangles.
Here a(n,m) = 0 for non-primitive Pythagorean triangles.
There is a one-to-one correspondence between the values n and m of this number triangle for which a(n,m) does not vanish and primitive solutions of x^2 + y^2 = z^2 with y even, namely x = n^2 - m^2, y = 2*n*m and z = n^2 + m^2. The mirror triangles with x even are not considered here. Therefore a(n,m) = (n^2 - m^2) + 2*n*m + (n^2 + m^2) = 2*n*(n+m) (for these solutions).
The number of non-vanishing entries in row n is A055034(n).
The sequence of the diagonal entries is 2*n*(2*n-1) = 2*A000384(n), n >= 2.
The ordered nonzero entries of this triangle gives A024364.
Note that all perimeters <= N will certainly be found if one consider all rows n = 2, 3, ..., floor((-1 + sqrt(2*N + 1))/2).
See also A070109(n) for the number of primitive Pythagorean triangles with perimeter n and leg y even.
REFERENCES
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991.
FORMULA
a(n,m) = 2*n*(n+m) if n > m >= 1, gcd(n,m) = 1, and n and m are integers of opposite parity (i.e., (-1)^{n+m} = -1), otherwise a(n,m) = 0.
EXAMPLE
The triangle a(n,m) begins:
n\m 1 2 3 4 5 6 7 8 9 10 11
2: 12
3: 0 30
4: 40 0 56
5: 0 70 0 90
6: 84 0 0 0 132
7: 0 126 0 154 0 182
8: 144 0 176 0 208 0 240
9: 0 198 0 234 0 0 0 306
10: 220 0 260 0 0 0 340 0 380
11: 0 286 0 330 0 374 0 418 0 462
12: 312 0 0 0 408 0 456 0 0 0 552
...
The primitive triangle for (n,m) = (2,1) is (x,y,z) = (3,4,5), therefore, a(2,1) = 3 + 4 + 5 = 12.
The primitive triangle for (n,m) = (7,4) is (x,y,z) = (33,56,65), therefore, a(7,4) = 33 + 56 + 65 = 154.
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, May 21 2013
STATUS
approved