Pairing 1 to 22 with differences of 1 or 11

In how many ways can the set $\{1,2,\dots,22\}$ be split into $11$ two-element blocks so that in every block the two numbers differ by exactly $1$ or exactly $11$?

$145$. Write $1$–$11$ above $12$–$22$ in a $2\times11$ grid: differences of $11$ are vertical neighbours and differences of $1$ are horizontal neighbours, plus one extra link between $11$ and $12$. Domino tilings of a $2\times n$ strip follow the Fibonacci recursion, giving $144$ for $n=11$. Pairing $\{11,12\}$ forces everything else, adding $1$ more way.