\(\DeclareMathOperator{\res}{Res}\)

This one’s another puzzle from work:

Consider a pigeon coop with \(n\) pigeonholes, arranged in a straight line. When a pigeon arrives at the coop, it will roost in a pigeonhole only if it is empty, and both neighboring pigeonholes are also empty. It selects such a pigeonhole uniformly at random, enters the pigeonhole, and does not leave. At some point, the coop will fill up, but not every pigeonhole will be occupied. What is the expected density of pigeons in the coop, as \(n\) grows large?

If you run a few simulations, you get that it’s about \(0.432332\ldots\). But this isn’t any easily recognizable number. What is it in closed form?