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

1,1

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 >= 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)

a(n) = n + 12 for n > 7. - Charles R Greathouse IV, May 26 2026

Example

9 is a term because 9 = 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, 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])
(PARI) a(n)=if(n>7, n+12, [6, 9, 12, 14, 15, 17, 18][n]) \\ Charles R Greathouse IV, May 26 2026

Crossrefs

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

Sequence in context: A185179 A072546 A387319 * A295670 A272466 A267918

Adjacent sequences: A344802 A344803 A344804 * A344806 A344807 A344808

Keyword

nonn,easy,changed

Author

David Consiglio, Jr., Jun 19 2021

Status

approved