A377123 Number of subsets of the first n nonzero triangular numbers whose sum is a nonzero triangular number.

1, 2, 4, 5, 8, 13, 18, 29, 48, 81, 136, 228, 420, 747, 1326, 2468, 4520, 8296, 15232, 28196, 52504, 96936, 181699, 340998, 640091, 1207386, 2282956, 4328536, 8211989, 15623003, 29787092, 56831499, 108591159, 207851120, 398222781, 763760138, 1466661416, 2819529654, 5425826011, 10450403468

Offset: 1

Author

Ilya Gutkovskiy, Oct 17 2024

Keywords

• nonn

Examples

			a(5) = 8 subsets: {1}, {3}, {6}, {10}, {15}, {6, 15}, {1, 3, 6} and {3, 10, 15}.
		

Crossrefs

Cf. A000217, A284250.

Programs

Python

from math import isqrt
from functools import cache
def cond(s): k = 8*s+1; return s > 0 and isqrt(k)**2 == k
def u(n): return n*(n+1)//2
@cache
def b(n, s):
    if n == 0: return int(cond(s))
    return b(n-1, s) + b(n-1, s+u(n))
a = lambda n: b(n, 0)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Oct 18 2024

Extensions

a(23) and beyond from Michael S. Branicky, Oct 18 2024