A292116 - OEIS

%I #23 Feb 09 2021 02:44:24

%S 13,14,33,34,61,62,97,98,103,141,142,193,194,253,254,321,322,397,398,

%T 481,482,573,574,673,674,713,781,782,897,898,1021,1022,1153,1154,1293,

%U 1294,1441,1442,1597,1598,1761,1762,1933,1934,2113,2114,2301,2302,2497,2498,2701,2702,2913,2914,3133,3134

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

%C It would be nice to have a more precise definition.

%C From _Ray Chandler_, Sep 11 2017: (Start)

%C The sequence is the union of three types of numbers:

%C (1) A060626 beginning with the 2nd term.

%C (2) A089508 beginning with the 3rd term and omitting even values (every third term).

%C (3) A082109 beginning with the 2nd term.

%C Note that there appear to be other solutions that are not covered by Theorem 12.

%C (End)

%H Ray Chandler, <a href="/A292116/b292116.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 lim=3000; a0={};

%t k=4; While[c=k^2-3;c<=lim,a0=Join[a0,{c,c+1}];k+=2];

%t k=2; While[c=Fibonacci[2k]*Fibonacci[2k+1]-1;c<=lim,If[OddQ[c],AppendTo[a0,c]];k++];

%t a0=Sort[a0] (* _Ray Chandler_, Sep 11 2017 *)

%Y Cf. A060626, A082109, A089508.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Sep 10 2017