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%I #15 Feb 20 2018 14:14:54
%S 21,36,56,253,496,561,1771,2926,3655,5985,26335,2895621,2919736,
%T 6471003,21474181,48792381,346700278,402073903,1260501229261,
%U 12864662659597529
%N Numbers n such that n and n-1 are both nontrivial binomial coefficients.
%C Nontrivial here means binomial(r,s) with 2 <= s <= r-2 (or the sequence would be uninteresting).
%C Blokhuis et al. show that the values given are complete up to 10^30, and conjecture that there are no more.
%H Aart Blokhuis, Andries Brouwer, Benne de Weger, <a href="http://math.colgate.edu/~integers/vol17.html">Binomial collisions and near collisions</a>, INTEGERS, Volume 17, Article A64, 2017 (also available as <a href="https://arxiv.org/abs/1707.06893">arXiv:1707.06893 [math.NT]</a>).
%e binomial(6,3)=20 and binomial(7,2)=binomial(7,5)=21 are the smallest adjacent pair, so a(1)=21.
%t nmax = 1000; t = Table[Binomial[n, k], {n, 4, nmax}, {k, 2, Floor[n/2]}] // Flatten // Sort // DeleteDuplicates; Select[Split[t, #2 == #1+1&], Length[#] > 1&][[All, 2]] (* _Jean-François Alcover_, Feb 20 2018 *)
%Y Cf. A003015.
%K nonn
%O 1,1
%A _Christopher E. Thompson_, Jan 19 2018