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%I #14 Aug 05 2016 04:57:33
%S 1,2,1,1,3,2,1,3,2,3,4,1,1,3,1,3,4,2,3,3,3,1,2,3,2,7,2,1,4,3,4,5,3,2,
%T 4,2,4,4,1,5,8,3,2,4,1,7,3,1,2,4,5,1,5,2,4,7,3,3,5,1,3,5,1,6,6,7,2,4,
%U 5,2,9,3,4,6,3,3,2,2,4,7
%N Number of ordered ways to write n as 4^k*(1+5*x^2+y^2) + z^2, where k,x,y,z are nonnegative integers with x <= y.
%C Conjecture: (i) a(n) > 0 for all n > 0.
%C (ii) Any positive integer can be written as 4^k*(1+5*x^2+y^2) + z^2, where k,x,y,z are nonnegative integers with y <= z.
%C See also A275656, A275675 and A275678 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A275676/b275676.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(4) = 1 since 4 = 4*(1+5*0^2+0^2) + 0^2 with 0 = 0.
%e a(259) = 1 since 259 = 4^0*(1+5*4^2+13^2) + 3^2 with 4 < 13.
%e a(333) = 1 since 333 = 4*(1+5*3^2+5^2) + 7^2 with 3 < 5.
%e a(621) = 1 since 621 = 4*(1+5*0^2+8^2) + 19^2 with 0 < 8.
%e a(717) = 1 since 717 = 4*(1+5*3^2+11^2) + 7^2 with 3 < 11.
%e a(1581) = 1 since 1581 = 4*(1+5*1^2+3^2) + 39^2 with 1 < 3.
%e a(2541) = 1 since 2541 = 4*(1+5*3^2+13^2) + 41^2 with 3 < 13.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t Do[r=0;Do[If[SQ[n-4^k*(1+5x^2+y^2)],r=r+1],{k,0,Log[4,n]},{x,0,Sqrt[(n/4^k-1)/6]},{y,x,Sqrt[n/4^k-1-5x^2]}];Print[n," ",r];Continue,{n,1,80}]
%Y Cf. A000118, A000290, A271518, A275648, A275656, A275675, A275678
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Aug 04 2016
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