A350861 Number of solutions to +-1^3 +- 2^3 +- 3^3 +- ... +- n^3 = 0 or 1.

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 4, 1, 4, 2, 6, 1, 4, 124, 12, 344, 536, 712, 1140, 713, 4574, 2260, 4384, 5956, 10634, 73758, 48774, 197767, 406032, 638830, 1147500, 1097442, 4249160, 3263500, 6499466, 11844316, 21907736, 82561050, 85185855, 261696060

Offset: 0

Author

Ilya Gutkovskiy, Jan 19 2022

Keywords

• nonn

Examples

			a(12) = 2: +1^3 + 2^3 - 3^3 + 4^3 - 5^3 - 6^3 - 7^3 + 8^3 + 9^3 - 10^3 - 11^3 + 12^3 = -1^3 - 2^3 + 3^3 - 4^3 + 5^3 + 6^3 + 7^3 - 8^3 - 9^3 + 10^3 + 11^3 - 12^3 = 0.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..99

Crossrefs

Cf. A000578, A025591, A113263, A158118.

Programs

Maple

f:= proc(n) local S,k,x,s;
  S:= mul(1 + x^(2*k^3),k=1..n);
  s:= sum(k^3,k=1..n);
  coeff(S,x,s) + coeff(S,x,s+1)
end proc:
map(f, [$0..50]); # Robert Israel, Mar 15 2023

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def b(n, i):
    if n > (i*(i+1)//2)**2: return 0
    if i == 0: return 1
    return b(n+i**3, i-1) + b(abs(n-i**3), i-1)
def a(n): return b(0, n) + b(1, n)
print([a(n) for n in range(46)]) # Michael S. Branicky, Jan 19 2022