This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A225952 #21 Dec 12 2015 16:49:22
%S A225952 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,
%T A225952 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,
%U A225952 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
%N A225952 Triangle read by rows, giving the even legs of primitive Pythagorean triangles, with zero entries for non-primitive triangles.
%C A225952 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.
%C A225952 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).
%C A225952 The sequence of the main diagonal is 2*n*(n-1) = 4*A000217 (n-1), n >= 2.
%C A225952 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).
%D A225952 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
%D A225952 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.
%H A225952 Kival Ngaokrajang, <a href="/A225952/a225952.pdf">Illustration of pattern of zero terms (non-isolated zeros are colored), for n = 1..103.</a>
%F A225952 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.
%e A225952 The triangle a(n,m) begins:
%e A225952 n\m 1 2 3 4 5 6 7 8 9 10 11 ...
%e A225952 2: 4
%e A225952 3: 0 12
%e A225952 4: 8 0 24
%e A225952 5: 0 20 0 40
%e A225952 6: 12 0 0 0 60
%e A225952 7: 0 28 0 56 0 84
%e A225952 8: 16 0 48 0 80 0 112
%e A225952 9: 0 36 0 72 0 0 0 144
%e A225952 10: 20 0 60 0 0 0 140 0 180
%e A225952 11: 0 44 0 88 0 132 0 176 0 220
%e A225952 12: 24 0 0 0 120 0 168 0 0 0 264
%e A225952 ...
%Y A225952 Cf. A222946 (hypotenuses), A225950 (odd legs), A225949 (leg sums), A225951 (perimeters), A120427 (even legs ordered), A008586 (multiples of 4).
%K A225952 nonn,easy,tabl
%O A225952 2,1
%A A225952 _Wolfdieter Lang_, May 23 2013
%E A225952 Edited. Refs. added. - _Wolfdieter Lang_, Jul 26 2014