Consider the strip $\{0,1,\ldots n\}\times\{0,1,2\}$ in $\mathbb{N}^2.$ Is a formula known for the total number of self avoiding walks in this strip starting at $(0,0)$ in terms of the parameter $n$?

*Edit:* I mean *all* the walks that take steps of $(\pm 1,0),(0,\pm1)$ as long as they are confined to that strip.

**Note:** Asked on math.stackexchange (see here) a week ago with no specific answer.