A348165 Number of solutions to +-1^2 +- 2^2 +- 3^2 +- ... +- n^2 = n.

1, 1, 0, 0, 1, 2, 0, 0, 2, 4, 0, 0, 19, 29, 0, 0, 127, 208, 0, 0, 1121, 1917, 0, 0, 10479, 19360, 0, 0, 113213, 204121, 0, 0, 1290968, 2363982, 0, 0, 15303057, 28397538, 0, 0, 187446097, 351339307, 0, 0, 2355979330, 4455357992, 0, 0, 30360404500, 57630025172

Offset: 0

Author

Ilya Gutkovskiy, Jan 28 2022

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A063890, A158092, A348892.

Programs

Maple

b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
      b(abs(n-i^2), i-1)+b(n+i^2, i-1))))((1+(3+2*i)*i)*i/6)
    end:
a:= n-> `if`(irem(n, 4)>1, 0, b(n$2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 28 2022

Mathematica

b[n_, i_] := b[n, i] = Function[m, If[n > m, 0, If[n == m, 1, b[Abs[n - i^2], i - 1] + b[n + i^2, i - 1]]]][(1 + (3 + 2*i)*i)*i/6];
a[n_] := If[Mod[n, 4] > 1, 0, b[n, n]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def b(n, i):
    if n > i*(i+1)*(2*i+1)//6: return 0
    if i == 0: return 1
    return b(n+i**2, i-1) + b(abs(n-i**2), i-1)
def a(n): return b(n, n)
print([a(n) for n in range(50)]) # Michael S. Branicky, Jan 28 2022

Formula

a(n) = [x^n] Product_{k=1..n} (x^(k^2) + 1/x^(k^2)).