A339569 Number of subsets of {1..n} whose cardinality is equal to the root-mean-square of the elements.

1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 5, 10, 16, 32, 56, 90, 134, 186, 304, 476, 746, 1308, 2522, 4845, 9129, 17260, 32684, 59908, 106181, 191779, 337793, 596689, 1061991, 1907311, 3518903, 6426672, 12093858, 22777645, 42886411, 81002076, 151575988, 285280108, 529313088

Offset: 1

Author

Ilya Gutkovskiy, Dec 09 2020

Keywords

• nonn

Examples

			a(12) = 10 subsets: {1}, {1, 2, 4, 5, 7, 11}, {1, 3, 5, 6, 8, 9}, {3, 4, 5, 6, 7, 9}, {1, 2, 3, 6, 7, 10, 12}, {2, 3, 4, 5, 8, 9, 12}, {2, 3, 6, 7, 8, 9, 10}, {3, 4, 5, 6, 7, 8, 12}, {1, 2, 5, 6, 9, 10, 11, 12} and {1, 4, 6, 7, 8, 9, 11, 12}.
		

Links

• Eric Weisstein's World of Mathematics, Root-Mean-Square

Crossrefs

Cf. A092670, A339454, A339484.

Programs

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def b(n, sos, c):
  if n == 0:
    if c>0:
      if sos==c*c*c: 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(1, 44)]) # Michael S. Branicky, Dec 10 2020

Extensions

a(24)-a(43) from Michael S. Branicky, Dec 09 2020