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A345508 Numbers that are the sum of ten squares in one or more ways.

%I A345508 #17 May 10 2024 08:52:11
%S A345508 10,13,16,18,19,21,22,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,
%T A345508 40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,
%U A345508 63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81
%N A345508 Numbers that are the sum of ten squares in one or more ways.
%H A345508 Sean A. Irvine, <a href="/A345508/b345508.txt">Table of n, a(n) for n = 1..1000</a>
%H A345508 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A345508 From _Chai Wah Wu_, May 09 2024: (Start)
%F A345508 All integers >= 24 are terms. Proof: since 5 can be written as the sum of 5 positive squares and any integer >= 34 can be written as a sum of 5 positive squares (see A025429), any integer >= 39 can be written as a sum of 10 positive squares. Integers from 24 to 38 are terms by inspection.
%F A345508 a(n) = 2*a(n-1) - a(n-2) for n > 9.
%F A345508 G.f.: x*(-x^8 + x^7 - x^6 + x^5 - x^4 - x^3 - 7*x + 10)/(x - 1)^2. (End)
%e A345508 13 is a term because 13 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2.
%o A345508 (Python)
%o A345508 from itertools import combinations_with_replacement as cwr
%o A345508 from collections import defaultdict
%o A345508 keep = defaultdict(lambda: 0)
%o A345508 power_terms = [x**2 for x in range(1, 1000)]
%o A345508 for pos in cwr(power_terms, 10):
%o A345508     tot = sum(pos)
%o A345508     keep[tot] += 1
%o A345508     rets = sorted([k for k, v in keep.items() if v >= 1])
%o A345508     for x in range(len(rets)):
%o A345508         print(rets[x])
%o A345508 (Python)
%o A345508 def A345508(n): return (10, 13, 16, 18, 19, 21, 22)[n-1] if n<8 else n+16 # _Chai Wah Wu_, May 09 2024
%Y A345508 Cf. A003333, A025429, A345498, A345509.
%K A345508 nonn
%O A345508 1,1
%A A345508 _David Consiglio, Jr._, Jun 19 2021