A225951 Triangle for perimeters of primitive Pythagorean triangles.

12, 0, 30, 40, 0, 56, 0, 70, 0, 90, 84, 0, 0, 0, 132, 0, 126, 0, 154, 0, 182, 144, 0, 176, 0, 208, 0, 240, 0, 198, 0, 234, 0, 0, 0, 306, 220, 0, 260, 0, 0, 0, 340, 0, 380, 0, 286, 0, 330, 0, 374, 0, 418, 0, 462, 312, 0, 0, 0, 408, 0, 456, 0, 0, 0, 552, 0, 390, 0, 442, 0, 494, 0, 546, 0, 598, 0, 650, 420, 0, 476, 0, 532, 0, 0, 0, 644, 0, 700, 0, 756

Offset: 2

Author

Wolfdieter Lang, May 21 2013

Keywords

• nonn
• easy
• tabl

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.

Examples

			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.
		

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.

Crossrefs

Cf. A024364 (nonzero, ordered), A225949 (leg sums), A222946 (hypotenuses), A000384 (half of the main diagonal), A070109.

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.