%I #5 Mar 04 2018 10:33:02
%S 1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,3,1,1,4,1,4,2,4,1,2,2,2,3,2,1,5,
%T 1,5,2,3,3,3,1,1,2,3,1,4,3,5,1,6,6,6,1,4,6,8,2,4,4,3,1,5,3,9,1,4,4,5,
%U 3,10,4,7,3,9,2,14,2,6,2,6,1
%N Number of ways to write n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x + 2*y and (z + 2*w)/3 are squares and w is even.
%C Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 7, 9, 14, 19, 22, 26, 34, 41, 4^k*m (k = 0,1,... and m = 1, 2, 3, 5, 10, 11, 13, 15).
%H Zhi-Wei Sun, <a href="/A300362/b300362.txt">Table of n, a(n) for n = 0..1000</a>
%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017-2018.
%e a(9) = 1 since 9^2 = 9^2 + 0^2 + 0^2 + 0^2 with 9 + 2*0 = 3^2 and 0 + 2*0 = 3*0^2.
%e a(13) = 1 since 13^2 = 4^2 + 0^2 + 3^2 + 12^2 with 4 + 2*0 = 2^2 and 3 + 2*12 = 3*3^2.
%e a(14) = 1 since 14^2 = 4^2 + 6^2 + 12^2 + 0^2 with 4 + 2*6 = 4^2 and 12 + 2*0 = 3*2^2.
%e a(15) = 1 since 15^2 = 9^2 + 0^2 + 12^2 + 0^2 with 9 + 2*0 = 3^2 and 12 + 2*0 = 3*2^2.
%e a(41) = 1 since 41 = 38^2 + 13^2 + 8^2 + 2^2 with 38 + 2*13 = 8^2 and 8 + 2*2 = 3*2^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={};Do[r=0;Do[If[SQ[x+2y]&&SQ[(n^2-x^2-y^2-z^2)/4]&&SQ[(z+2*Sqrt[n^2-x^2-y^2-z^2])/3],r=r+1],{x,0,n},{y,0,Sqrt[n^2-x^2]},{z,0,Sqrt[n^2-x^2-y^2]}];tab=Append[tab,r],{n,0,80}];Print[tab]
%Y Cf. A000118, A000290, A271518, A281976, A299924, A300219, A300360.
%K nonn
%O 0,7
%A _Zhi-Wei Sun_, Mar 04 2018