How many paths are there between $(0,0)$ and $(n,n)$ if you include all eight common cardinal directions: North, East, South, West, Northeast, Northwest, Southeast, and Southwest. The only condition is if $(i,j)$ is in the path, $0≤i≤n$ and $0≤j≤n$, and if $(k,l)$ is also in the same path, both $i ≠ k$ and $j ≠ l$.
We would have
$(0,0) => (0,1) => (1,1)$
$(0,0) => (1,0) => (1,1)$
$(0,0) => (1,1)$
$(0,0) => (0,1) => (1,0) => (1,1)$
$(0,0) => (1,0) => (0,1) => (1,1)$
So $a(1) = 5.$
(A good link is https://oeis.org/A007764 -- these are the number of paths with no diagonal movement.)
Are there any suggestions on how to approach this or other links with information? Thanks in advance!