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A345482 Numbers that are the sum of seven squares in five or more ways.

%I A345482 #14 May 10 2024 01:37:19
%S A345482 45,54,55,57,58,60,61,63,64,66,67,69,70,71,72,73,74,75,76,77,78,79,80,
%T A345482 81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,
%U A345482 103,104,105,106,107,108,109,110,111,112,113,114,115,116,117
%N A345482 Numbers that are the sum of seven squares in five or more ways.
%H A345482 Sean A. Irvine, <a href="/A345482/b345482.txt">Table of n, a(n) for n = 1..1000</a>
%F A345482 Conjectures from _Chai Wah Wu_, Apr 25 2024: (Start)
%F A345482 a(n) = 2*a(n-1) - a(n-2) for n > 13.
%F A345482 G.f.: x*(-x^12 + x^11 - x^10 + x^9 - x^8 + x^7 - x^6 + x^5 - x^4 + x^3 - 8*x^2 - 36*x + 45)/(x - 1)^2. (End)
%e A345482 54 = 1^2 + 1^2 + 1^2 + 1^2 + 3^2 + 4^2 + 5^2
%e A345482    = 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 6^2
%e A345482    = 1^2 + 1^2 + 3^2 + 3^2 + 3^2 + 3^2 + 4^2
%e A345482    = 1^2 + 2^2 + 2^2 + 2^2 + 3^2 + 4^2 + 4^2
%e A345482    = 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 5^2
%e A345482 so 54 is a term.
%o A345482 (Python)
%o A345482 from itertools import combinations_with_replacement as cwr
%o A345482 from collections import defaultdict
%o A345482 keep = defaultdict(lambda: 0)
%o A345482 power_terms = [x**2 for x in range(1, 1000)]
%o A345482 for pos in cwr(power_terms, 7):
%o A345482     tot = sum(pos)
%o A345482     keep[tot] += 1
%o A345482     rets = sorted([k for k, v in keep.items() if v >= 5])
%o A345482     for x in range(len(rets)):
%o A345482         print(rets[x])
%Y A345482 Cf. A344809, A345481, A345483, A345492, A345523.
%K A345482 nonn
%O A345482 1,1
%A A345482 _David Consiglio, Jr._, Jun 20 2021