A345498 Numbers that are the sum of nine squares in one or more ways.

9, 12, 15, 17, 18, 20, 21, 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, 78, 79, 80

Offset

1,1

Links

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

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

All integers >= 23 are terms. See A345508 for 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 - 6*x + 9)/(x - 1)^2. (End)

a(n) = n + 15 for n > 7. - Charles R Greathouse IV, May 26 2026

Example

12 is a term because 12 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2.

Prog

(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, 9):
    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])
(PARI) a(n)=if(n>7, n+15, [9, 12, 15, 17, 18, 20, 21][n]) \\ Charles R Greathouse IV, May 26 2026

Crossrefs

Cf. A003332, A345488, A345499, A345508.

Sequence in context: A377873 A369002 A369976 * A248903 A138945 A119486

Adjacent sequences: A345495 A345496 A345497 * A345499 A345500 A345501

Keyword

nonn,easy

Author

David Consiglio, Jr., Jun 19 2021

Status

approved