A378170 Number of subsets of the first n nonzero tetrahedral numbers whose sum is a nonzero tetrahedral number.

1, 2, 3, 5, 7, 8, 11, 13, 19, 34, 45, 72, 113, 171, 262, 388, 638, 1128, 1928, 3370, 5584, 9691, 17129, 30493, 54785, 94510, 169817, 308491, 559176, 1019487, 1816043, 3333698, 6153695, 11384025, 21100254, 38262081, 71096456, 132675454, 247900732, 463959984

Offset: 1

Author

Ilya Gutkovskiy, Nov 18 2024

Keywords

• nonn

Examples

			a(8) = 13 subsets: {1}, {4}, {10}, {20}, {35}, {56}, {84}, {120}, {1, 20, 35}, {1, 35, 84}, {10, 35, 120}, {1, 4, 10, 20} and {1, 4, 20, 56, 84}.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..103

Crossrefs

Cf. A000292, A377123, A378171.

Programs

Python

from sympy import integer_nthroot
def is_tetra(n): return (c:=integer_nthroot(6*n, 3)[0])*(c+1)*(c+2) == 6*n
from functools import cache
@cache
def b(n, s):
    if n == 0:
        if s > 0 and is_tetra(s): return 1
        return 0
    return b(n-1, s) + b(n-1, s+n*(n+1)*(n+2)//6)
a = lambda n: b(n, 0)
print([a(n) for n in range(1, 30)]) # Michael S. Branicky, Nov 18 2024

Extensions

a(24) and beyond from Michael S. Branicky, Nov 18 2024