A275676 Number of ordered ways to write n as 4^k*(1+5*x^2+y^2) + z^2, where k,x,y,z are nonnegative integers with x <= y.

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

Offset: 1

Author

Zhi-Wei Sun, Aug 04 2016

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 0.

(ii) Any positive integer can be written as 4^k*(1+5*x^2+y^2) + z^2, where k,x,y,z are nonnegative integers with y <= z.

See also A275656, A275675 and A275678 for similar conjectures.

Examples

			a(4) = 1 since 4 = 4*(1+5*0^2+0^2) + 0^2 with 0 = 0.
a(259) = 1 since 259 = 4^0*(1+5*4^2+13^2) + 3^2 with 4 < 13.
a(333) = 1 since 333 = 4*(1+5*3^2+5^2) + 7^2 with 3 < 5.
a(621) = 1 since 621 = 4*(1+5*0^2+8^2) + 19^2 with 0 < 8.
a(717) = 1 since 717 = 4*(1+5*3^2+11^2) + 7^2 with 3 < 11.
a(1581) = 1 since 1581 = 4*(1+5*1^2+3^2) + 39^2 with 1 < 3.
a(2541) = 1 since 2541 = 4*(1+5*3^2+13^2) + 41^2 with 3 < 13.
		

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, A275678

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-4^k*(1+5x^2+y^2)],r=r+1],{k,0,Log[4,n]},{x,0,Sqrt[(n/4^k-1)/6]},{y,x,Sqrt[n/4^k-1-5x^2]}];Print[n," ",r];Continue,{n,1,80}]