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A275471 Number of ordered ways to write n as 4^k*(1+x^2+y^2)+z^2, where k,x,y,z are nonnegative integers with x <= y and x == y (mod 2).

%I A275471 #18 Aug 11 2016 09:12:33
%S A275471 1,1,1,2,3,1,1,1,2,2,1,3,3,1,1,2,3,2,2,5,5,1,1,1,3,2,2,4,2,2,1,1,2,2,
%T A275471 2,5,6,1,2,2,4,3,1,3,5,2,1,3,2,2,3,7,5,2,3,1,4,2,1,6,2,2,2,2,4,3,3,5,
%U A275471 8,2,1,2,6,2,3,6,4,2,1,5
%N A275471 Number of ordered ways to write n as 4^k*(1+x^2+y^2)+z^2, where k,x,y,z are nonnegative integers with x <= y and x == y (mod 2).
%C A275471 Conjecture: a(n) > 0 except for n = 449.
%C A275471 See also A275656, A275678 and A275738 for related conjectures.
%C A275471 As x^2 + y^2 = 2*((x+y)/2)^2 + 2*((x-y)/2)^2, we see that {x^2 + y^2: x and y are integers with x == y (mod 2)} = {2*x^2 + 2*y^2: x and y are integers}.
%H A275471 Zhi-Wei Sun, <a href="/A275471/b275471.txt">Table of n, a(n) for n = 1..10000</a>
%H A275471 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 A275471 a(8) = 1 since 8 = 4*(1+0^2+0^2) + 2^2 with 0+0 even.
%e A275471 a(31) = 1 since 31 = 4^0*(1+1^2+5^2) + 2^2 with 1+5 even.
%e A275471 a(47) = 1 since 47 = 4^0*(1+1^2+3^2) + 6^2 with 1+3 even.
%e A275471 a(79) = 1 since 79 = 4^0*(1+5^2+7^2)+2^2 with 5+7 even.
%e A275471 a(1009) = 1 since 1009 = 4^2*(1+1^2+1^2) + 31^2 with 1+1 even.
%e A275471 a(7793) = 1 since 7793 = 4^2*(1+12^2+18^2) + 17^2 with 12+18 even.
%t A275471 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A275471 Do[r=0;Do[If[SQ[n-4^k*(1+2x^2+2y^2)],r=r+1],{k,0,Log[4,n]},{x,0,Sqrt[(n/4^k-1)/4]},{y,x,Sqrt[(n/4^k-1-2x^2)/2]}];Print[n," ",r];Continue,{n,1,80}]
%Y A275471 Cf. A000118, A000290, A271518, A275648, A275656, A275675, A275676, A275678, A275738.
%K A275471 nonn
%O A275471 1,4
%A A275471 _Zhi-Wei Sun_, Aug 11 2016