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
Keywords
Examples
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.
Links
- Sean A. Irvine, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (2,-1).
Programs
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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
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)