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%I #6 Feb 18 2017 14:28:08
%S 1,2,2,2,4,3,5,1,3,4,2,2,2,5,7,3,2,5,3,1,4,7,7,3,2,2,2,4,3,8,8,3,2,2,
%T 4,4,9,3,9,3,4,5,6,3,3,7,5,2,2,11,6,5,4,7,7,4,2,4,3,2,2,5,10,6,4,5,9,
%U 1,7,8,10,4,4,5,6,5,3,9,3,2,3
%N Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2*x - y and 4*z^2 + 724*z*w + w^2 are squares.
%C Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 7, 19, 67, 191, 235, 265, 347, 888, 2559).
%C By the linked JNT paper, any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z*(z-w) = 0. Whether z = 0 or z = w, the number 4*z^2 + 724*z*w + w^2 is definitely a square.
%C See also A282562 for a similar conjecture.
%H Zhi-Wei Sun, <a href="/A282561/b282561.txt">Table of n, a(n) for n = 0..10000</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.
%e a(7) = 1 since 7 = 1^2 + 2^2 + 1^2 + 1^2 with 2*1 - 2 = 0^2 and 4*1^2 + 724*1*1 + 1^2 = 27^2.
%e a(19) = 1 since 19 = 1^2 + 1^2 + 1^2 + 4^2 with 2*1 - 1 = 1^2 and 4*1^2 + 724*1*4 + 4^2 = 54^2.
%e a(67) = 1 since 67 = 4^2 + 7^2 + 1^2 + 1^2 with 2*4 - 7 = 1^2 and 4*1^2 + 724*1*1 + 1^2 = 27^2.
%e a(191) = 1 since 191 = 9^2 + 2^2 + 5^2 + 9^2 with 2*9 - 2 = 4^2 and 4*5^2 + 724*5*9 + 9^2 = 181^2.
%e a(235) = 1 since 235 = 7^2 + 13^2 + 1^2 + 4^2 with 2*7 - 13 = 1^2 and 4*1^2 + 724*1*4 + 4^2 = 54^2.
%e a(265) = 1 since 265 = 4^2 + 7^2 + 10^2 + 10^2 with 2*4 - 7 = 1 and 4*10^2 + 724*10*10 + 10^2 = 270^2.
%e a(347) = 1 since 347 = 8^2 + 7^2 + 15^2 + 3^2 with 2*8 - 7 = 3^2 and 4*15^2 + 724*15*3 + 3^2 = 183^2.
%e a(888) = 1 since 888 = 14^2 + 12^2 + 8^2 + 22^2 with 2*14 - 12 = 4^2 and 4*8^2 + 724*8*22 + 22^2 = 358^2.
%e a(2559) = 1 since 2559 = 26^2 + 3^2 + 5^2 + 43^2 with 2*26 - 3 = 7^2 and 4*5^2 + 724*5*43 + 43^2 = 397^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t Do[r=0;Do[If[SQ[2x-y],Do[If[SQ[n-x^2-y^2-z^2]&&SQ[4z^2+724z*Sqrt[n-x^2-y^2-z^2]+(n-x^2-y^2-z^2)],r=r+1],{z,0,Sqrt[n-x^2-y^2]}]],{y,0,Sqrt[4n/5]},{x,Ceiling[y/2],Sqrt[n-y^2]}];Print[n," ",r];Continue,{n,0,80}]
%Y Cf. A000118, A000290, A271518, A281976, A282463, A282494, A282495, A282545, A282562.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Feb 18 2017
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