A344805 Numbers that are the sum of six squares in one or more ways.

6, 9, 12, 14, 15, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Author

David Consiglio, Jr., Jun 19 2021

Keywords

• nonn

Examples

			9 is a term because 9 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2.
		

Links

• Sean A. Irvine, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A003329, A047700, A344806, A345478, A345508.

Programs

Python

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**2 for x in range(1, 1000)]
for pos in cwr(power_terms, 6):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 1])
    for x in range(len(rets)):
        print(rets[x])

Formula

From Chai Wah Wu, Jun 12 2025: (Start)

All integers >= 20 are terms. See A345508 for a similar proof.

a(n) = 2*a(n-1) - a(n-2) for n > 9.

G.f.: x*(-x^8 + x^7 - x^6 + x^5 - x^4 - x^3 - 3*x + 6)/(x - 1)^2. (End)