A357356 Number of nonempty subsets of {1..n} whose elements have an even average.

0, 1, 3, 4, 6, 13, 25, 38, 64, 119, 219, 384, 686, 1285, 2405, 4428, 8236, 15569, 29463, 55582, 105326, 200847, 383609, 732744, 1403004, 2694391, 5181663, 9973416, 19226166, 37125547, 71770069, 138871244, 269005540, 521666967, 1012555015, 1966957674, 3824166974

Offset: 1

Author

Ilya Gutkovskiy, Sep 25 2022

Keywords

• nonn

Examples

			a(6) = 13 subsets: {2}, {4}, {6}, {1, 3}, {2, 6}, {3, 5}, {1, 2, 3}, {1, 5, 6}, {2, 4, 6}, {3, 4, 5}, {1, 4, 5, 6}, {2, 3, 5, 6} and {2, 3, 4, 5, 6}.
		

Crossrefs

Cf. A051293, A309160, A357355.

Programs

Python

from functools import lru_cache
def cond(s, c): q, r = divmod(s, c); return r == 0 and q&1 == 0
@lru_cache(maxsize=None)
def b(n, s, c):
    if n == 0: return int (c > 0 and cond(s, c))
    return b(n-1, s, c) + b(n-1, s+n, c+1)
a = lambda n: b(n, 0, 0)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Sep 25 2022

Formula

a(n) = A051293(n) - A357355(n). - Michael S. Branicky, Sep 25 2022

Extensions

a(24)-a(37) from Michael S. Branicky, Sep 25 2022