A225951 - OEIS

%I #13 Oct 25 2016 06:25:14

%S 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,

%T 208,0,240,0,198,0,234,0,0,0,306,220,0,260,0,0,0,340,0,380,0,286,0,

%U 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

%N Triangle for perimeters of primitive Pythagorean triangles.

%C See the Hardy-Wright (Theorem 225, p. 190) and Niven-Zuckerman-Montgomery (Theorem 5.5, p. 232) references for primitive Pythagorean triangles.

%C Here a(n,m) = 0 for non-primitive Pythagorean triangles.

%C 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).

%C The number of non-vanishing entries in row n is A055034(n).

%C The sequence of the diagonal entries is 2*n*(2*n-1) = 2*A000384(n), n >= 2.

%C The ordered nonzero entries of this triangle gives A024364.

%C   Note that all perimeters <= N will certainly be found if one consider all rows n = 2, 3, ..., floor((-1 + sqrt(2*N + 1))/2).

%C See also A070109(n) for the number of primitive Pythagorean triangles with perimeter n and leg y even.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.

%D 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.

%F 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.

%e The triangle a(n,m) begins:

%e n\m   1    2   3    4    5    6    7    8    9   10   11

%e 2:   12

%e 3:    0   30

%e 4:   40    0  56

%e 5:    0   70   0   90

%e 6:   84    0   0    0  132

%e 7:    0  126   0  154    0  182

%e 8:  144    0 176    0  208    0  240

%e 9:    0  198   0  234    0    0    0  306

%e 10: 220    0 260    0    0    0  340    0  380

%e 11:   0  286   0  330    0  374    0  418    0  462

%e 12: 312    0   0    0  408    0  456    0    0    0  552

%e ...

%e The primitive triangle for (n,m) = (2,1) is (x,y,z) = (3,4,5), therefore, a(2,1) = 3 + 4 + 5 = 12.

%e The primitive triangle for (n,m) = (7,4) is (x,y,z) = (33,56,65), therefore, a(7,4) = 33 + 56 + 65 = 154.

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

%K nonn,easy,tabl

%O 2,1

%A _Wolfdieter Lang_, May 21 2013