A275471 Number of ordered ways to write n as 4^k*(1+x^2+y^2)+z^2, where k,x,y,z are nonnegative integers with x <= y and x == y (mod 2).

1, 1, 1, 2, 3, 1, 1, 1, 2, 2, 1, 3, 3, 1, 1, 2, 3, 2, 2, 5, 5, 1, 1, 1, 3, 2, 2, 4, 2, 2, 1, 1, 2, 2, 2, 5, 6, 1, 2, 2, 4, 3, 1, 3, 5, 2, 1, 3, 2, 2, 3, 7, 5, 2, 3, 1, 4, 2, 1, 6, 2, 2, 2, 2, 4, 3, 3, 5, 8, 2, 1, 2, 6, 2, 3, 6, 4, 2, 1, 5

Offset: 1

Author

Zhi-Wei Sun, Aug 11 2016

Keywords

• nonn

Comments

Conjecture: a(n) > 0 except for n = 449.

See also A275656, A275678 and A275738 for related conjectures.

As x^2 + y^2 = 2*((x+y)/2)^2 + 2*((x-y)/2)^2, we see that {x^2 + y^2: x and y are integers with x == y (mod 2)} = {2*x^2 + 2*y^2: x and y are integers}.

Examples

			a(8) = 1 since 8 = 4*(1+0^2+0^2) + 2^2 with 0+0 even.
a(31) = 1 since 31 = 4^0*(1+1^2+5^2) + 2^2 with 1+5 even.
a(47) = 1 since 47 = 4^0*(1+1^2+3^2) + 6^2 with 1+3 even.
a(79) = 1 since 79 = 4^0*(1+5^2+7^2)+2^2 with 5+7 even.
a(1009) = 1 since 1009 = 4^2*(1+1^2+1^2) + 31^2 with 1+1 even.
a(7793) = 1 since 7793 = 4^2*(1+12^2+18^2) + 17^2 with 12+18 even.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Crossrefs

Cf. A000118, A000290, A271518, A275648, A275656, A275675, A275676, A275678, A275738.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-4^k*(1+2x^2+2y^2)],r=r+1],{k,0,Log[4,n]},{x,0,Sqrt[(n/4^k-1)/4]},{y,x,Sqrt[(n/4^k-1-2x^2)/2]}];Print[n," ",r];Continue,{n,1,80}]