For what values of $n$ is $n^2+n+2$ a power of $2$?

Working on isometric paths in hypercubes, I came up with the following simple, yet (imo) interesting problem. For what natural numbers $n$ exists a natural number $t$ such that $n^2+n+2=2^t$? The first few terms are $n=0,1,2,5,90$, and these are all below one million. Does someone have any idea how to approach this problem? I basically only want to know whether there exist infinitely many $n$ or not (maybe even that 0, 1, 2, 5, and 90 are the only possible ones).

Thanks,

Sacha

That there are no more positive solutions was proved by Nagell, settling a conjecture of Ramanujan. The problem is discussed in this paper, and in this Wikipedia article.

To see that these solve the problem, a small preliminary transformation of your equation is useful. Rewrite it as $4n^2+4n+8=2^k$, and then as $(2n+1)^2+7=2^k$. We have arrived at the Ramanujan-Nagell equation.