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%I #14 Sep 29 2020 04:06:57
%S 5,6,12,24,27,30,39,48,57,60,71,85,86,90,96,106,111,113,119,120,123,
%T 126,135,159,172,180,192,212,225,240,249,252,263,287,293,294,297,306,
%U 329,344,347,350,360,363,365,378,384,402,424,427,429,437,438,447,449,479
%N Numbers k such that binomial(2*k,k) cannot be represented as the sum of three squares.
%C Granville and Zhu show that the density of these numbers is 1/8.
%H Amiram Eldar, <a href="/A099473/b099473.txt">Table of n, a(n) for n = 1..10000</a>
%H Andrew Granville and Yiliang Zhu, <a href="http://jstor.org/stable/2323831">Representing binomial coefficients as sums of squares</a>, Amer. Math. Monthly, Vol. 97, No. 6 (1990), pp. 486-493; <a href="http://www.dms.umontreal.ca/~andrew/PDF/YZhu.pdf">alternative link</a>.
%t NoRepAs3Sqrs[n_] := Module[{e2}, e2=IntegerExponent[n, 2]; If[EvenQ[e2], 7==Mod[n/2^e2, 8], False]]; Select[Range[500], NoRepAs3Sqrs[Binomial[2#, # ]]&]
%Y Cf. A004215 (sums of 4 but no fewer nonzero squares), A099472.
%K nonn
%O 1,1
%A _T. D. Noe_, Oct 18 2004