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%I #19 Aug 05 2016 04:57:03
%S 1,1,1,1,1,2,1,2,2,2,2,1,1,2,1,2,2,1,2,1,3,2,2,3,2,4,1,1,2,2,3,3,1,1,
%T 2,2,3,3,1,3,3,2,1,2,1,5,3,2,2,3,4,1,4,2,3,5,2,2,3,1,3,3,1,4,2,4,1,2,
%U 3,2,6,2,3,3,2,2,2,2,2,3
%N Number of ordered ways to write n as 4^k*(1+x^2+y^2)+5*z^2, where k,x,y,z are nonnegative integers with x <= y.
%C Conjecture: a(n) > 0 for all n > 0.
%C This is stronger than the known fact that any natural number can be written as w^2 + x^2 + y^2 + 5*z^2 with w,x,y,z integers.
%C See also A275656, A275676 and A275678 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A275675/b275675.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723 [math.GM], 2016.
%e a(43) = 1 since 43 = 4^0*(1+1^2+6^2) + 5*1^2.
%e a(45) = 1 since 45 = 4*(1+0^2+3^2) + 5*1^2.
%e a(237) = 1 since 237 = 4^3*(1+1^2+1^2) + 5*3^2.
%e a(561) = 1 since 561 = 4*(1+8^2+8^2) + 5*3^2.
%e a(9777) = 1 since 9777 = 4*(1+11^2+31^2) + 5*33^2.
%e a(39108) = 1 since 39108 = 4^2*(1+11^2+31^2) + 5*66^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t Do[r=0;Do[If[SQ[(n-4^k*(1+x^2+y^2))/5],r=r+1],{k,0,Log[4,n]},{x,0,Sqrt[(n/4^k-1)/2]},{y,x,Sqrt[n/4^k-1-x^2]}];Print[n," ",r];Continue,{n,1,80}]
%Y Cf. A000118, A000290, A271518, A275648, A275656, A275676, A275678.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Aug 04 2016
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