A345510 Numbers that are the sum of ten squares in three or more ways.

34, 37, 40, 42, 43, 45, 46, 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, 97, 98, 99, 100, 101, 102, 103, 104

Offset: 1

Author

David Consiglio, Jr., Jun 20 2021

Keywords

• nonn

Examples

			37 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 5^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 3^2 + 3^2 + 3^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 4^2
   = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2
so 37 is a term.
		

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. A345500, A345509, A345551.

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, 10):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 3])
    for x in range(len(rets)):
        print(rets[x])

Formula

From Chai Wah Wu, May 09 2024: (Start)

All integers >= 48 are terms. Proof: since 29 can be written as the sum of 5 positive squares in 3 ways and any integer >= 34 can be written as a sum of 5 positive squares (see A025429), any integer >= 63 can be written as a sum of 10 positive squares in 3 or more ways. Integers from 48 to 62 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 - 31*x + 34)/(x - 1)^2. (End)