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%I #49 Feb 19 2017 14:00:01
%S 0,1,2,1,2,4,2,1,4,5,2,2,4,3,1,1,5,5,3,4,6,6,2,4,5,5,5,3,5,5,3,1,7,8,
%T 2,5,6,7,2,5,7,4,2,3,7,4,3,2,5,7,5,6,7,12,4,3,7,7,2,1,7,5,4,1,7,7,3,7,
%U 8,6,2,5,7,6,4,4,8,4,1,4
%N Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,z positive integers and y,w nonnegative integers such that both 9*x^2 + 246*x*y + y^2 and 9*z^2 + 666*z*w + w^2 are squares.
%C The first three values of n with a(n) = 0 are 1, 214635, 241483.
%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 x*(x-y) = 0, Whether x = 0 or x = y, both 9*x^2 + 246*x*y + y^2 and 9*x^2 + 666*x*y + y^2 are squares.
%H Zhi-Wei Sun, <a href="/A281729/b281729.txt">Table of n, a(n) for n = 1..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(2) = 1 since 2 = 1^2 + 0^2 + 1^2 + 0^2 with 9*1^2 + 246*1*0 + 1^2 = 3^2 and 9*1^2 + 666*1*0 + 0^2 = 3^2.
%e a(4) = 1 since 4 = 1^2 + 1^2 + 1^2 + 1^2 with 9*1^2 + 246*1*1 + 1^2 = 16^2 and 9*1^2 + 666*1*1 + 1^2 = 26^2.
%e a(15) = 1 since 15 = 2^2 + 3^2 + 1^2 + 1^2 with 9*2^2 + 246*2*3 + 3^2 = 39^2 and 9*1^2 + 666*1*1 + 1^2 = 26^2.
%e a(159) = 1 since 159 = 11^2 + 3^2 + 5^2 + 2^2 with 9*11^2 + 246*11*3 + 3^2 = 96^2 and 9*5^2 + 666*5*2 + 2^2 = 83^2.
%e a(515) = 1 since 515 = 15^2 + 0^2 + 17^2 + 1^2 with 9*15^2 + 246*15*0 + 0^2 = 45^2 and 9*17^2 + 666*17*1 + 1^2 = 118^2.
%e a(9795) = 1 since 9795 = 35^2 + 91^2 + 17^2 + 0^2 with 9*35^2 + 246*35*91 + 91^2 = 896^2 and 9*17^2 + 666*17*0 + 0^2 = 51^2.
%e a(84155) = 1 since 84155 = 281^2 + 0^2 + 35^2 + 63^2 with 9*281^2 + 246*281*0 + 0^2 = 843^2 and 9*35^2 + 666*35*63 + 63^2 = 1218^2.
%e a(121003) = 1 since 121003 = 319^2 + 87^2 + 3^2 + 108^2 with 9*319^2 + 246*319*87 + 87^2 = 2784^2 and 9*3^2 + 666*3*108 + 108^2 = 477^2.
%e a(133647) = 1 since 133647 = 217^2 + 217^2 + 115^2 + 162^2 with 9*217^2 + 246*217*217 + 217^2 = 3472^2 and 9*115^2 + 666*115*162 + 162^2 = 3543^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t Do[r=0;Do[If[SQ[9x^2+246x*y+y^2],Do[If[SQ[n-x^2-y^2-z^2]&&SQ[9z^2+666z*Sqrt[n-x^2-y^2-z^2]+(n-x^2-y^2-z^2)],r=r+1],{z,1,Sqrt[n-x^2-y^2]}]],{x,1,Sqrt[n]},{y,0,Sqrt[n-x^2]}];Print[n," ",r];Continue,{n,1,80}]
%Y Cf. A000118, A000290, A271518, A281976, A282561, A282562.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Feb 19 2017
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