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A292116 Numbers for which there exists a nontrivial bisection of binomial coefficients as given by Theorem 12 of Ionascu et al. (2016).

%I A292116 #23 Feb 09 2021 02:44:24
%S A292116 13,14,33,34,61,62,97,98,103,141,142,193,194,253,254,321,322,397,398,
%T A292116 481,482,573,574,673,674,713,781,782,897,898,1021,1022,1153,1154,1293,
%U A292116 1294,1441,1442,1597,1598,1761,1762,1933,1934,2113,2114,2301,2302,2497,2498,2701,2702,2913,2914,3133,3134
%N A292116 Numbers for which there exists a nontrivial bisection of binomial coefficients as given by Theorem 12 of Ionascu et al. (2016).
%C A292116 It would be nice to have a more precise definition.
%C A292116 From _Ray Chandler_, Sep 11 2017: (Start)
%C A292116 The sequence is the union of three types of numbers:
%C A292116 (1) A060626 beginning with the 2nd term.
%C A292116 (2) A089508 beginning with the 3rd term and omitting even values (every third term).
%C A292116 (3) A082109 beginning with the 2nd term.
%C A292116 Note that there appear to be other solutions that are not covered by Theorem 12.
%C A292116 (End)
%H A292116 Ray Chandler, <a href="/A292116/b292116.txt">Table of n, a(n) for n = 1..10000</a>
%H A292116 Eugen J. Ionascu, Thor Martinsen, and Pantelimon Stanica, <a href="https://arxiv.org/abs/1610.02063">Bisecting binomial coefficients</a>, arXiv:1610.02063 [math.CO], 2016. See p. 18.
%t A292116 lim=3000; a0={};
%t A292116 k=4; While[c=k^2-3;c<=lim,a0=Join[a0,{c,c+1}];k+=2];
%t A292116 k=2; While[c=Fibonacci[2k]*Fibonacci[2k+1]-1;c<=lim,If[OddQ[c],AppendTo[a0,c]];k++];
%t A292116 a0=Sort[a0] (* _Ray Chandler_, Sep 11 2017 *)
%Y A292116 Cf. A060626, A082109, A089508.
%K A292116 nonn
%O A292116 1,1
%A A292116 _N. J. A. Sloane_, Sep 10 2017