OFFSET
2,1
COMMENTS
For primitive Pythagorean triples (x,y,z) see the Niven et al. reference, Theorem 5.5, p. 232, and the Hardy-Wright reference, Theorem 225, p. 190.
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 (for these solutions).
The number of non-vanishing entries in row n is A055034(n).
The sequence of the main diagonal is 2*n -1 = A005408(n-1),
n >= 2.
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.
LINKS
FORMULA
a(n,m) = n^2 - m^2 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 12 ...
2: 3
3: 0 5
4: 15 0 7
5: 0 21 0 9
6: 35 0 0 0 11
7: 0 45 0 33 0 13
8: 63 0 55 0 39 0 15
9: 0 77 0 65 0 0 0 17
10: 99 0 91 0 0 0 51 0 19
11: 0 117 0 105 0 85 0 57 0 21
12: 143 0 0 0 119 0 95 0 0 0 23
13: 0 165 0 153 0 133 0 105 0 69 0 25
...
a(6,1) = 35 from the primitive triangle (35,12,37).
a(6,2) = 0 because n and m are even (not allowed n, m values for primitive triangles).
a(6,3) = 0 because gcd(6,3) = 3 (not 1, hence not allowed).
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, May 23 2013
STATUS
approved