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A345498 Numbers that are the sum of nine squares in one or more ways.

%I A345498 #9 Jun 12 2025 13:38:03
%S A345498 9,12,15,17,18,20,21,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,
%T A345498 39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,
%U A345498 62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80
%N A345498 Numbers that are the sum of nine squares in one or more ways.
%H A345498 Sean A. Irvine, <a href="/A345498/b345498.txt">Table of n, a(n) for n = 1..1000</a>
%H A345498 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A345498 From _Chai Wah Wu_, Jun 12 2025: (Start)
%F A345498 All integers >= 23 are terms. See A345508 for similar proof.
%F A345498 a(n) = 2*a(n-1) - a(n-2) for n > 9.
%F A345498 G.f.: x*(-x^8 + x^7 - x^6 + x^5 - x^4 - x^3 - 6*x + 9)/(x - 1)^2. (End)
%e A345498 12 is a term because 12 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2.
%o A345498 (Python)
%o A345498 from itertools import combinations_with_replacement as cwr
%o A345498 from collections import defaultdict
%o A345498 keep = defaultdict(lambda: 0)
%o A345498 power_terms = [x**2 for x in range(1, 1000)]
%o A345498 for pos in cwr(power_terms, 9):
%o A345498     tot = sum(pos)
%o A345498     keep[tot] += 1
%o A345498     rets = sorted([k for k, v in keep.items() if v >= 1])
%o A345498     for x in range(len(rets)):
%o A345498         print(rets[x])
%Y A345498 Cf. A003332, A345488, A345499, A345508.
%K A345498 nonn
%O A345498 1,1
%A A345498 _David Consiglio, Jr._, Jun 19 2021