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 A225950 #17 Dec 12 2015 16:53:11
%S A225950 3,0,5,15,0,7,0,21,0,9,35,0,0,0,11,0,45,0,33,0,13,63,0,55,0,39,0,15,0,
%T A225950 77,0,65,0,0,0,17,99,0,91,0,0,0,51,0,19,0,117,0,105,0,85,0,57,0,21,
%U A225950 143,0,0,0,119,0,95,0,0,0,23,0,165,0,153,0,133,0,105,0,69,0,25,195,0,187,0,171,0,0,0,115,0,75,0,27,0,221,0,209,0,0,0,161,0,0,0,0,0,29
%N A225950 Triangle for odd legs of primitive Pythagorean triangles.
%C A225950 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 A225950 Here a(n,m) = 0 for non-primitive Pythagorean triangles.
%C A225950 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 (for these solutions).
%C A225950 The number of non-vanishing entries in row n is A055034(n).
%C A225950 The sequence of the main diagonal is 2*n -1 = A005408(n-1),
%C A225950 n >= 2.
%C A225950 If the zeros are eliminated and the numbers are sorted nondecreasingly (multiple entries appear) one obtains A120890. All odd numbers >= 3 appear, they are given in A005408. Note that all odd legs x will be found if one takes in the triangle n = 2, ..., floor((x+1)/2).
%D A225950 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
%D A225950 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 A225950 Kival Ngaokrajang, <a href="/A225950/a225950.pdf">Illustration of pattern of zero terms (non-isolated zeros are colored), for n = 1..50.</a>
%F A225950 a(n,m) = n^2 - m^2 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 A225950 The triangle a(n,m) begins:
%e A225950 n\m 1 2 3 4 5 6 7 8 9 10 11 12 ...
%e A225950 2: 3
%e A225950 3: 0 5
%e A225950 4: 15 0 7
%e A225950 5: 0 21 0 9
%e A225950 6: 35 0 0 0 11
%e A225950 7: 0 45 0 33 0 13
%e A225950 8: 63 0 55 0 39 0 15
%e A225950 9: 0 77 0 65 0 0 0 17
%e A225950 10: 99 0 91 0 0 0 51 0 19
%e A225950 11: 0 117 0 105 0 85 0 57 0 21
%e A225950 12: 143 0 0 0 119 0 95 0 0 0 23
%e A225950 13: 0 165 0 153 0 133 0 105 0 69 0 25
%e A225950 ...
%e A225950 a(6,1) = 35 from the primitive triangle (35,12,37).
%e A225950 a(6,2) = 0 because n and m are even (not allowed n, m values for primitive triangles).
%e A225950 a(6,3) = 0 because gcd(6,3) = 3 (not 1, hence not allowed).
%Y A225950 Cf. A222946 (hypotenuses), A225952 (even legs), A225949 (leg sums), A225951 (perimeters), A120890 (odd legs, ordered).
%K A225950 nonn,easy,tabl
%O A225950 2,1
%A A225950 _Wolfdieter Lang_, May 23 2013