A345478 Numbers that are the sum of seven squares in one or more ways.

7, 10, 13, 15, 16, 18, 19, 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, 78

Offset: 1

Author

David Consiglio, Jr., Jun 19 2021

Keywords

• nonn

Examples

			10 is a term because 10 = 1^2 + 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. A003330, A344805, A345479, A345488, A345508.

Programs

Mathematica

ssQ[n_]:=Count[IntegerPartitions[n,{7}],?(AllTrue[Sqrt[#],IntegerQ]&)]>0; Select[ Range[ 80],ssQ] (* _Harvey P. Dale, Jun 22 2022 *)

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, 7):
    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 >= 21 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 - 4*x + 7)/(x - 1)^2. (End)