A345509 Numbers that are the sum of ten squares in two or more ways.

25, 28, 31, 33, 34, 36, 37, 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, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96

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, May 09 2024: (Start)

All integers >= 39 are terms. Proof: since 20 can be written as the sum of 5 positive squares in 2 ways and any integer >= 34 can be written as a sum of 5 positive squares (see A025429), any integer >= 54 can be written as a sum of 10 positive squares in 2 or more ways. Integers from 39 to 53 are terms by inspection.

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 - 22*x + 25)/(x - 1)^2. (End)

a(n) = n + 31 for n > 7. - Charles R Greathouse IV, May 22 2026

Example

28 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 4^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2
so 28 is a term.

Mathematica

LinearRecurrence[{2, -1}, {25, 28, 31, 33, 34, 36, 37, 39, 40}, 100] (* Paolo Xausa, May 22 2026 *)

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, 10):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 2])
    for x in range(len(rets)):
        print(rets[x])
(Python)
def A345509(n): return (25, 28, 31, 33, 34, 36, 37)[n-1] if n<8 else n+31 # Chai Wah Wu, May 09 2024
(PARI) a(n)=if(n>7, n+31, [25, 28, 31, 33, 34, 36, 37][n]) \\ Charles R Greathouse IV, May 22 2026

Crossrefs

Cf. A025429, A345499, A345508, A345510, A345550.

Sequence in context: A015734 A221617 A239521 * A295748 A254226 A195613

Adjacent sequences: A345506 A345507 A345508 * A345510 A345511 A345512

Keyword

nonn,easy

Author

David Consiglio, Jr., Jun 20 2021

Status

approved