A345508 Numbers that are the sum of ten squares in one or more ways.

10, 13, 16, 18, 19, 21, 22, 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, 81

Offset: 1

Author

David Consiglio, Jr., Jun 19 2021

Keywords

• nonn

Examples

			13 is a term because 13 = 1^2 + 1^2 + 1^2 + 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. A003333, A025429, A345498, A345509.

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 >= 1])
    for x in range(len(rets)):
        print(rets[x])

Python

def A345508(n): return (10, 13, 16, 18, 19, 21, 22)[n-1] if n<8 else n+16 # Chai Wah Wu, May 09 2024

Formula

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

All integers >= 24 are terms. Proof: since 5 can be written as the sum of 5 positive squares and any integer >= 34 can be written as a sum of 5 positive squares (see A025429), any integer >= 39 can be written as a sum of 10 positive squares. Integers from 24 to 38 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 - 7*x + 10)/(x - 1)^2. (End)