A225952 Triangle read by rows, giving the even legs of primitive Pythagorean triangles, with zero entries for non-primitive triangles.

4, 0, 12, 8, 0, 24, 0, 20, 0, 40, 12, 0, 0, 0, 60, 0, 28, 0, 56, 0, 84, 16, 0, 48, 0, 80, 0, 112, 0, 36, 0, 72, 0, 0, 0, 144, 20, 0, 60, 0, 0, 0, 140, 0, 180, 0, 44, 0, 88, 0, 132, 0, 176, 0, 220, 24, 0, 0, 0, 120, 0, 168, 0, 0, 0, 264, 0, 52, 0, 104, 0, 156, 0, 208, 0, 260, 0, 312, 28, 0, 84, 0, 140, 0, 0, 0, 252, 0, 308, 0, 364

Offset: 2

Author

Wolfdieter Lang, May 23 2013

Keywords

• nonn
• easy
• tabl

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.

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) = 2*n*m (for these solutions). The number of non-vanishing entries in row n is A055034(n).

The sequence of the main diagonal is 2*n*(n-1) = 4*A000217 (n-1), n >= 2.

If the 0 entries are eliminated and the numbers are ordered nondecreasingly (multiple entries appear) the sequence becomes A120427. All its entries are positive integer multiples of 4, shown in A008586(n), n >= 1. Note that all even legs <= N are certainly reached if one considers in the triangle rows n = 2, ..., floor(N/2).

Examples

			The triangle a(n,m) begins:
n\m   1   2   3   4    5    6    7    8    9    10   11 ...
2:    4
3:    0  12
4:    8   0  24
5:    0  20   0  40
6:   12   0   0   0   60
7:    0  28   0  56    0   84
8:   16   0  48   0   80    0  112
9:    0  36   0  72    0    0    0  144
10:  20   0  60   0    0    0  140    0  180
11:   0  44   0  88    0  132    0  176    0   220
12:  24   0   0   0  120    0  168    0    0     0  264
...
		

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

• Kival Ngaokrajang, Illustration of pattern of zero terms (non-isolated zeros are colored), for n = 1..103.

Crossrefs

Cf. A222946 (hypotenuses), A225950 (odd legs), A225949 (leg sums), A225951 (perimeters), A120427 (even legs ordered), A008586 (multiples of 4).

Formula

a(n,m) = 2*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.

Extensions

Edited. Refs. added. - Wolfdieter Lang, Jul 26 2014