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%I A225951 #13 Oct 25 2016 06:25:14
%S A225951 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 A225951 208,0,240,0,198,0,234,0,0,0,306,220,0,260,0,0,0,340,0,380,0,286,0,
%U A225951 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 A225951 Triangle for perimeters of primitive Pythagorean triangles.
%C A225951 See the Hardy-Wright (Theorem 225, p. 190) and Niven-Zuckerman-Montgomery (Theorem 5.5, p. 232) references for primitive Pythagorean triangles.
%C A225951 Here a(n,m) = 0 for non-primitive Pythagorean triangles.
%C A225951 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 A225951 The number of non-vanishing entries in row n is A055034(n).
%C A225951 The sequence of the diagonal entries is 2*n*(2*n-1) = 2*A000384(n), n >= 2.
%C A225951 The ordered nonzero entries of this triangle gives A024364.
%C A225951 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 A225951 See also A070109(n) for the number of primitive Pythagorean triangles with perimeter n and leg y even.
%D A225951 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
%D A225951 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 A225951 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 A225951 The triangle a(n,m) begins:
%e A225951 n\m 1 2 3 4 5 6 7 8 9 10 11
%e A225951 2: 12
%e A225951 3: 0 30
%e A225951 4: 40 0 56
%e A225951 5: 0 70 0 90
%e A225951 6: 84 0 0 0 132
%e A225951 7: 0 126 0 154 0 182
%e A225951 8: 144 0 176 0 208 0 240
%e A225951 9: 0 198 0 234 0 0 0 306
%e A225951 10: 220 0 260 0 0 0 340 0 380
%e A225951 11: 0 286 0 330 0 374 0 418 0 462
%e A225951 12: 312 0 0 0 408 0 456 0 0 0 552
%e A225951 ...
%e A225951 The primitive triangle for (n,m) = (2,1) is (x,y,z) = (3,4,5), therefore, a(2,1) = 3 + 4 + 5 = 12.
%e A225951 The primitive triangle for (n,m) = (7,4) is (x,y,z) = (33,56,65), therefore, a(7,4) = 33 + 56 + 65 = 154.
%Y A225951 Cf. A024364 (nonzero, ordered), A225949 (leg sums), A222946 (hypotenuses), A000384 (half of the main diagonal), A070109.
%K A225951 nonn,easy,tabl
%O A225951 2,1
%A A225951 _Wolfdieter Lang_, May 21 2013