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Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x + 3*y + 5*z and (at least) one of y,z,w are squares.
5

%I #10 Feb 18 2017 14:26:48

%S 1,2,2,2,2,1,1,1,1,3,3,2,1,2,4,2,2,4,5,3,2,2,2,2,1,5,5,2,1,5,8,1,2,3,

%T 3,3,2,3,5,5,2,8,5,1,1,6,6,1,2,5,9,5,4,2,5,5,2,5,4,5,2,1,5,3,2,7,9,5,

%U 2,3,6,2,2,8,9,5,3,5,9,2,1

%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 x + 3*y + 5*z and (at least) one of y,z,w are squares.

%C Conjecture: a(n) > 0 for all n = 0,1,2,....

%C This is stronger than the 1-3-5 conjecture (cf. A271518).

%C By the linked JNT paper, any nonnegative integer can be expressed as the sum of a fourth power and three squares.

%H Zhi-Wei Sun, <a href="/A282542/b282542.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(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1 + 3*1 + 5*1 = 3^2 and 1 = 1^2.

%e a(28) = 1 since 28 = 1^2 + 1^2 + 1^2 + 5^2 with 1 + 3*1 + 5*1 = 3^2 and 1 = 1^2.

%e a(47) = 1 since 47 = 3^2 + 1^2 + 6^2 + 1^2 with 3 + 3*1 + 5*6 = 6^2 and 1 = 1^2.

%e a(92) = 1 since 92 = 1^2 + 1^2 + 9^2 + 3^2 with 1 + 3*1 + 5*1 = 3^2 and 9 = 3^2.

%e a(188) = 1 since 188 = 7^2 + 9^2 + 3^2 + 7^2 with 7 + 3*9 + 5*3 = 7^2 and 9 = 3^2.

%e a(248) = 1 since 248 = 10^2 + 2^2 + 0^2 + 12^2 with 10 + 3*2 + 5*0 = 4^2 and 0 = 0^2.

%e a(388) = 1 since 388 = 13^2 + 1^2 + 13^2 + 7^2 with 13 + 3*1 + 5*13 = 9^2 and 1 = 1^2.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

%t Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&(SQ[y]||SQ[z]||SQ[Sqrt[n-x^2-y^2-z^2]])&&SQ[x+3y+5z],r=r+1],{x,0,n^(1/2)},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,0,80}]

%Y Cf. A000118, A000290, A270969, A271518, A281976.

%K nonn

%O 0,2

%A _Zhi-Wei Sun_, Feb 17 2017