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A344805 Numbers that are the sum of six squares in one or more ways.

%I A344805 #23 Jun 12 2025 13:38:17
%S A344805 6,9,12,14,15,17,18,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,
%T A344805 36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,
%U A344805 59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77
%N A344805 Numbers that are the sum of six squares in one or more ways.
%H A344805 Sean A. Irvine, <a href="/A344805/b344805.txt">Table of n, a(n) for n = 1..1000</a>
%H A344805 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A344805 From _Chai Wah Wu_, Jun 12 2025: (Start)
%F A344805 All integers >= 20 are terms. See A345508 for a similar proof.
%F A344805 a(n) = 2*a(n-1) - a(n-2) for n > 9.
%F A344805 G.f.: x*(-x^8 + x^7 - x^6 + x^5 - x^4 - x^3 - 3*x + 6)/(x - 1)^2. (End)
%e A344805 9 is a term because 9 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2.
%o A344805 (Python)
%o A344805 from itertools import combinations_with_replacement as cwr
%o A344805 from collections import defaultdict
%o A344805 keep = defaultdict(lambda: 0)
%o A344805 power_terms = [x**2 for x in range(1, 1000)]
%o A344805 for pos in cwr(power_terms, 6):
%o A344805     tot = sum(pos)
%o A344805     keep[tot] += 1
%o A344805     rets = sorted([k for k, v in keep.items() if v >= 1])
%o A344805     for x in range(len(rets)):
%o A344805         print(rets[x])
%Y A344805 Cf. A003329, A047700, A344806, A345478, A345508.
%K A344805 nonn
%O A344805 1,1
%A A344805 _David Consiglio, Jr._, Jun 19 2021