A336815 Number of subsets of {1..n} whose sum of squares of elements is a square.

1, 2, 3, 4, 6, 7, 10, 12, 17, 26, 37, 69, 120, 233, 417, 781, 1386, 2561, 4638, 8387, 15495, 27709, 51580, 94054, 176266, 330004, 618846, 1174439, 2216002, 4232301, 8041866, 15344759, 29258898, 55850376, 106792759, 204203789, 391147474, 749434144, 1439261966

Offset: 0

Author

Ilya Gutkovskiy, Dec 09 2020

Keywords

• nonn

Examples

			a(8) = 17 subsets: {}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {3, 4}, {6, 8}, {1, 4, 8}, {2, 3, 6}, {2, 4, 5, 6}, {1, 2, 4, 6, 8}, {1, 3, 4, 5, 7} and {2, 4, 6, 7, 8}.
		

Crossrefs

Cf. A000290, A126024, A339612.

Programs

Python

from sympy.ntheory.primetest import is_square
from functools import lru_cache
@lru_cache(maxsize=None)
def b(n, sos, c):
  if n == 0:
    if is_square(sos): return 1
    return 0
  return b(n-1, sos, c) + b(n-1, sos+n*n, c+1)
a = lambda n: b(n, 0, 0)
print([a(n) for n in range(40)]) # Michael S. Branicky, Dec 10 2020

Formula

a(n) = 1 + Sum_{k=1..n} A339612(k).

Extensions

a(24)-a(38) from Michael S. Branicky, Dec 09 2020