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%I #7 Aug 05 2016 05:14:48
%S 1,2,1,1,3,3,1,2,3,4,2,1,2,3,2,1,4,4,1,3,5,3,1,3,5,5,3,1,2,7,2,2,5,3,
%T 3,3,6,2,2,4,6,7,1,2,4,7,1,1,3,5,5,2,5,5,4,3,8,4,2,2,1,7,3,1,6,8,2,4,
%U 8,6,2,4,6,3,4,1,3,6,2,3
%N Number of ordered ways to write n as 4^k*(1+4*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+4*x^2+y^2) + z^2, where k,x,y,z are nonnegative integers with x <= z.
%C This is stronger than Lagrange's four-square theorem. We have shown that each n = 1,2,3,... can be written as 4^k*(1+4*x^2+y^2) + z^2 with k,x,y,z nonnegative integers.
%C See also A275656, A275675 and A275676 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A275678/b275678.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(12) = 1 since 12 = 4*(1+4*0^2+1^2) + 2^2 with 0 < 1.
%e a(19) = 1 since 19 = 4^0*(1+4*0^2+3^2) + 3^2 with 0 < 3.
%e a(61) = 1 since 61 = 4*(1+4*1^2+2^2) + 5^2 with 1 < 2.
%e a(125) = 1 since 125 = 4*(1+4*0^2+0^2) + 11^2 with 0 = 0.
%e a(359) = 1 since 359 = 4^0*(1+4*7^2+9^2) + 9^2 with 7 < 9.
%e a(196253) = 1 since 196253 = 4*(1+4*0^2+0^2) + 443^2 with 0 = 0.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t Do[r=0;Do[If[SQ[n-4^k*(1+4x^2+y^2)],r=r+1],{k,0,Log[4,n]},{x,0,Sqrt[(n/4^k-1)/5]},{y,x,Sqrt[n/4^k-1-4x^2]}];Print[n," ",r];Continue,{n,1,80}]
%Y Cf. A000118, A000290, A271518, A275648, A275656, A275675, A275676.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Aug 05 2016
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