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A345476 Numbers that are the sum of six squares in nine or more ways.

%I A345476 #15 May 10 2024 01:39:59
%S A345476 78,81,84,86,89,92,93,95,99,100,101,102,104,105,107,108,110,111,113,
%T A345476 114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,
%U A345476 131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148
%N A345476 Numbers that are the sum of six squares in nine or more ways.
%H A345476 Sean A. Irvine, <a href="/A345476/b345476.txt">Table of n, a(n) for n = 1..1000</a>
%F A345476 Conjectures from _Chai Wah Wu_, Jan 05 2024: (Start)
%F A345476 a(n) = 2*a(n-1) - a(n-2) for n > 20.
%F A345476 G.f.: x*(-x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - 3*x^9 + 2*x^8 + x^7 - 2*x^6 + x^4 - x^3 - 75*x + 78)/(x - 1)^2. (End)
%e A345476 81 = 1^2 + 1^2 + 1^2 + 2^2 + 5^2 + 7^2
%e A345476    = 1^2 + 1^2 + 2^2 + 5^2 + 5^2 + 5^2
%e A345476    = 1^2 + 1^2 + 3^2 + 3^2 + 5^2 + 6^2
%e A345476    = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 8^2
%e A345476    = 1^2 + 2^2 + 3^2 + 3^2 + 3^2 + 7^2
%e A345476    = 1^2 + 4^2 + 4^2 + 4^2 + 4^2 + 4^2
%e A345476    = 2^2 + 2^2 + 2^2 + 2^2 + 4^2 + 7^2
%e A345476    = 2^2 + 2^2 + 4^2 + 4^2 + 4^2 + 5^2
%e A345476    = 2^2 + 3^2 + 3^2 + 3^2 + 5^2 + 5^2
%e A345476    = 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 6^2
%e A345476 so 81 is a term.
%o A345476 (Python)
%o A345476 from itertools import combinations_with_replacement as cwr
%o A345476 from collections import defaultdict
%o A345476 keep = defaultdict(lambda: 0)
%o A345476 power_terms = [x**2 for x in range(1, 1000)]
%o A345476 for pos in cwr(power_terms, 6):
%o A345476     tot = sum(pos)
%o A345476     keep[tot] += 1
%o A345476     rets = sorted([k for k, v in keep.items() if v >= 9])
%o A345476     for x in range(len(rets)):
%o A345476         print(rets[x])
%Y A345476 Cf. A344802, A344812, A345477, A345486, A345518.
%K A345476 nonn
%O A345476 1,1
%A A345476 _David Consiglio, Jr._, Jun 20 2021