A339484 Number of subsets of {1..n} whose cardinality is equal to the average of the elements.

1, 1, 2, 3, 4, 7, 11, 17, 28, 47, 80, 139, 245, 436, 784, 1419, 2585, 4738, 8729, 16154, 30015, 55966, 104682, 196378, 369384, 696494, 1316252, 2492683, 4729673, 8990374, 17118020, 32644544, 62345875, 119235519, 228333179, 437790086, 840362539, 1614894770, 3106516468

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..300

Eric W. Weisstein's World of Mathematics, Arithmetic Mean

Example

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

Prog

(Python)
from itertools import combinations
def a(n):
    ss, s = 0, range(1, n+1)
    for r in range(1, n+1):
        rr = r*r
        ss += sum(sum(subs)==rr for subs in combinations(s, r))
    return ss
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Dec 06 2020
(Python)
from functools import lru_cache
from itertools import combinations
@lru_cache(maxsize=None)
def A339484(n):
    return 1 if n == 1 else A339484(n-1)+sum(sum(d)+n==(i+1)**2 for i in range(1, n) for d in combinations(range(1, n), i)) # Chai Wah Wu, Dec 07 2020
(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def b(n, s, c):
    if n == 0: return c and int(s == c*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, 101)]) # Michael S. Branicky, Oct 06 2022

Crossrefs

Cf. A051293, A126024.

Sequence in context: A303025 A346020 A192669 * A072164 A060987 A359743

Adjacent sequences: A339481 A339482 A339483 * A339485 A339486 A339487

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 06 2020

Extensions

a(24)-a(32) from Michael S. Branicky, Dec 06 2020

a(33)-a(35) from Chai Wah Wu, Dec 07 2020

a(36)-a(39) from Michael S. Branicky, Dec 08 2020

Status

approved