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

1,4

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}.

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A252168 A100619 A211984 * A094006 A208879 A179617

Adjacent sequences: A275468 A275469 A275470 * A275472 A275473 A275474

Keyword

nonn

Author

Zhi-Wei Sun, Aug 11 2016

Status

approved