A345486 - OEIS

%I #10 Jan 06 2024 09:23:26

%S 69,70,78,79,81,82,85,87,88,90,91,93,94,95,96,97,98,99,100,101,102,

%T 103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,

%U 120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136

%N Numbers that are the sum of seven squares in nine or more ways.

%H Sean A. Irvine, <a href="/A345486/b345486.txt">Table of n, a(n) for n = 1..1000</a>

%F Conjectures from _Chai Wah Wu_, Jan 05 2024: (Start)

%F a(n) = 2*a(n-1) - a(n-2) for n > 13.

%F G.f.: x*(-x^12 + x^11 - x^10 + x^9 - x^8 - x^7 + 2*x^6 - x^5 + x^4 - 7*x^3 + 7*x^2 - 68*x + 69)/(x - 1)^2. (End)

%e 70 is a term because 70 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 8^2 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 7^2 = 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 5^2 + 5^2 = 1^2 + 1^2 + 2^2 + 4^2 + 4^2 + 4^2 + 4^2 = 1^2 + 1^2 + 3^2 + 3^2 + 3^2 + 4^2 + 5^2 = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 7^2 = 1^2 + 2^2 + 2^2 + 2^2 + 4^2 + 4^2 + 5^2 = 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 5^2 + 5^2 = 2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 3^2 + 6^2 = 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 4^2.

%o (Python)

%o from itertools import combinations_with_replacement as cwr

%o from collections import defaultdict

%o keep = defaultdict(lambda: 0)

%o power_terms = [x**2 for x in range(1, 1000)]

%o for pos in cwr(power_terms, 7):

%o     tot = sum(pos)

%o     keep[tot] += 1

%o     rets = sorted([k for k, v in keep.items() if v >= 9])

%o     for x in range(len(rets)):

%o         print(rets[x])

%Y Cf. A345476, A345485, A345487, A345496, A345527.

%K nonn

%O 1,1

%A _David Consiglio, Jr._, Jun 20 2021