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A345509 Numbers that are the sum of ten squares in two or more ways.

%I A345509 #12 May 10 2024 08:51:51
%S A345509 25,28,31,33,34,36,37,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,
%T A345509 55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,
%U A345509 78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96
%N A345509 Numbers that are the sum of ten squares in two or more ways.
%H A345509 Sean A. Irvine, <a href="/A345509/b345509.txt">Table of n, a(n) for n = 1..1000</a>
%H A345509 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A345509 From _Chai Wah Wu_, May 09 2024: (Start)
%F A345509 All integers >= 39 are terms. Proof: since 20 can be written as the sum of 5 positive squares in 2 ways and any integer >= 34 can be written as a sum of 5 positive squares (see A025429), any integer >= 54 can be written as a sum of 10 positive squares in 2 or more ways. Integers from 39 to 53 are terms by inspection.
%F A345509 a(n) = 2*a(n-1) - a(n-2) for n > 9.
%F A345509 G.f.: x*(-x^8 + x^7 - x^6 + x^5 - x^4 - x^3 - 22*x + 25)/(x - 1)^2. (End)
%e A345509 28 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 4^2
%e A345509    = 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2
%e A345509 so 28 is a term.
%o A345509 (Python)
%o A345509 from itertools import combinations_with_replacement as cwr
%o A345509 from collections import defaultdict
%o A345509 keep = defaultdict(lambda: 0)
%o A345509 power_terms = [x**2 for x in range(1, 1000)]
%o A345509 for pos in cwr(power_terms, 10):
%o A345509     tot = sum(pos)
%o A345509     keep[tot] += 1
%o A345509     rets = sorted([k for k, v in keep.items() if v >= 2])
%o A345509     for x in range(len(rets)):
%o A345509         print(rets[x])
%o A345509 (Python)
%o A345509 def A345509(n): return (25, 28, 31, 33, 34, 36, 37)[n-1] if n<8 else n+31 # _Chai Wah Wu_, May 09 2024
%Y A345509 Cf. A025429, A345499, A345508, A345510, A345550.
%K A345509 nonn
%O A345509 1,1
%A A345509 _David Consiglio, Jr._, Jun 20 2021