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A345478 Numbers that are the sum of seven squares in one or more ways.

%I A345478 #13 Jun 12 2025 13:38:30
%S A345478 7,10,13,15,16,18,19,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,
%T A345478 37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,
%U A345478 60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78
%N A345478 Numbers that are the sum of seven squares in one or more ways.
%H A345478 Sean A. Irvine, <a href="/A345478/b345478.txt">Table of n, a(n) for n = 1..1000</a>
%H A345478 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A345478 From _Chai Wah Wu_, Jun 12 2025: (Start)
%F A345478 All integers >= 21 are terms. See A345508 for a similar proof.
%F A345478 a(n) = 2*a(n-1) - a(n-2) for n > 9.
%F A345478 G.f.: x*(-x^8 + x^7 - x^6 + x^5 - x^4 - x^3 - 4*x + 7)/(x - 1)^2. (End)
%e A345478 10 is a term because 10 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2.
%t A345478 ssQ[n_]:=Count[IntegerPartitions[n,{7}],_?(AllTrue[Sqrt[#],IntegerQ]&)]>0; Select[ Range[ 80],ssQ] (* _Harvey P. Dale_, Jun 22 2022 *)
%o A345478 (Python)
%o A345478 from itertools import combinations_with_replacement as cwr
%o A345478 from collections import defaultdict
%o A345478 keep = defaultdict(lambda: 0)
%o A345478 power_terms = [x**2 for x in range(1, 1000)]
%o A345478 for pos in cwr(power_terms, 7):
%o A345478     tot = sum(pos)
%o A345478     keep[tot] += 1
%o A345478     rets = sorted([k for k, v in keep.items() if v >= 1])
%o A345478     for x in range(len(rets)):
%o A345478         print(rets[x])
%Y A345478 Cf. A003330, A344805, A345479, A345488, A345508.
%K A345478 nonn
%O A345478 1,1
%A A345478 _David Consiglio, Jr._, Jun 19 2021