is it known whether or not there are infinitely many pairs of consecutive terms in this sequence: http://oeis.org/A006987 ?
The sequence is the list of numbers expressible in the form $\dbinom{n}{k}$, where $1<k<n-1$, and $n,k\in \mathbb{N}$ (obviously).
I doubt if it is known -- indeed, I would be surprised if you can say with certainty whether there are infinitely many solutions $m,n\in\mathbb{N}$ to $$ {{n}\choose{2}}={{m}\choose{3}}+1. $$ It is easy to generate a large number of terms in the given sequence. I generated all terms up to $10^{13}$ (there are $4517489$ of them). The following is the list of consecutive pairs, where
X (N, K)indicates that $X={{N}\choose{K}}$. Though the entries become sparser, the lack of an obvious pattern and the large size of the final entry suggest it would be hard to prove either that the sequence terminates or that it goes on forever.