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%I #17 Mar 02 2017 11:04:07
%S 1,2,2,1,2,6,5,2,1,4,5,1,1,2,3,2,2,6,9,3,5,8,7,6,2,4,2,2,1,5,7,6,2,8,
%T 10,3,5,5,4,4,1,1,8,1,2,6,7,4,1,7,12,3,5,5,7,10,2,7,9,1,2,8,8,12,2,11,
%U 13,2,7,11,9,8,3,4,10,3,2,6,6,10,6
%N Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 2*y and z + 2*w both squares, where x and w are nonnegative integers, and y and z are integers.
%C Conjecture: (i) a(n) > 0 for all n = 0,1,2,....
%C (ii) Any nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x + 3*y and z + 3*w both squares, where x,y,z are integers and w is a nonnegative integer.
%C (iii) Every nonnegative integer can be expressed as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that both x + 2*y and z + 3*w are squares.
%C (vi) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w an integer such that |2*x-y| is a square and |2*z-w| is twice a square. Also, each nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w an integer such that |2*x-y| is twice a square and |2*z-w| is a square.
%C (v) Every n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that |(x-2*y)*(z-2*w)| is twice a square. Also, any positive integer n can be written as x^2 + y^2 + z^2 + w^2 with x a positive integer and y,z,w nonnegative integers such that (2*x+y)*(2*z-w) is twice a square.
%C (vi) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that both x + 2*y and z^2 - w^2 (or z^2 + 8*w^2, or 7*z^2 + 9*w^2) are squares.
%C (vii) Any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2*x - y and 64*z^2 - 84*z*w + 21*w^2 (or 81*z^2 - 112*z*w + 56*w^2) are squares.
%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 integers such that x + 2*y is a square.
%H Zhi-Wei Sun, <a href="/A283170/b283170.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.
%F a(3) = 1 since 3 = 1^2 + 0^2 + (-1)^2 + 1^2 with 1 + 2*0 = 1^2 and (-1)+2*1 = 1^2.
%F a(4) = 2 since 4 = 0^2 + 0^2 + 0^2 + 2^2 with 0 + 2*0 = 0^2 and 0 + 2*2 = 2^2, and also 4 = 0^2 + 2^2 + 0^2 + 0^2 with 0 + 2*2 = 2^2 and 0 + 2*0 = 0^2.
%F a(8) = 1 since 8 = 0^2 + 2^2 + 0^2 + 2^2 with 0 + 2*2 = 2^2 and 0 + 2*2 = 2^2.
%F a(11) = 1 since 11 = 3^2 + (-1)^2 + 1^2 + 0^2 with 3 + 2*(-1) = 1^2 and 1 + 2*0 = 1^2.
%F a(12) = 1 since 12 = 3^2 + (-1)^2 + (-1)^2 + 1^2 with 3 + 2*(-1) = 1^2 and (-1) + 2*1 = 1^2.
%F a(28) = 1 since 28 = 3^2 + (-1)^2 + 3^2 + 3^2 with 3 + 2*(-1) = 1^2 and 3 + 2*3 = 3^2.
%F a(40) = 1 since 40 = 4^2 + (-2)^2 + (-4)^2 + 2^2 with 4 + 2*(-2) = 0^2 and (-4) + 2*2 = 0^2.
%F a(41) = 1 since 41 = 6^2 + (-1)^2 + 0^2 + 2^2 with 6 + 2*(-1) = 2^2 and 0 + 2*2 = 2^2.
%F a(332) = 1 since 332 = 11^2 + 7^2 + (-9)^2 + 9^2 with 11 + 2*7 = 5^2 and (-9) + 2*9 = 3^2.
%F a(443) = 1 since 443 = 19^2 + (-9)^2 + 1^2 + 0^2 with 19 + 2*(-9) = 1^2 and 1 + 2*0 = 1^2.
%F a(488) = 1 since 488 = 12^2 + 2^2 + (-12)^2 + 14^2 with 12 + 2*2 = 4^2 and (-12) + 2*14 = 4^2.
%F a(808) = 1 since 808 = 8^2 + 14^2 + (-8)^2 + 22^2 with 8 + 2*14 = 6^2 and (-8) + 2*22 = 6^2.
%F a(892) = 1 since 892 = 27^2 + (-1)^2 + (-9)^2 + 9^2 with 27 + 2*(-1) = 5^2 and (-9) + 2*9 = 3^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t Do[r=0;Do[If[SQ[x+2(-1)^j*y],Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(-1)^k*z+2*Sqrt[n-x^2-y^2-z^2]],r=r+1],{z,0,Sqrt[n-x^2-y^2]},{k,0,Min[z,1]}]],{x,0,Sqrt[n]},{y,0,Sqrt[n-x^2]},{j,0,Min[y,1]}];Print[n," ",r];Continue,{n,0,80}]
%Y Cf. A000118, A000290, A271518, A281976, A281939, A282561, A282562.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Mar 02 2017
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