# Seeking to make sense of OEIS A360444

Having come across an interesting OEIS comment about two pairs of ants on a grid (https://oeis.org/A360444) I've spent some time trying to deduce how these counts were obtained. My hope is to gather notes on the actual routes that fulfill the conditions. It's proven difficult. The author hasn't responded to an email and I wonder if their claims are actually right. The history section (https://oeis.org/history?seq=A360444) leaves some clues as to more details that aren't available on the surface.

I list here what I think I've managed to grasp. Since they say a $$3 x 3$$ grid has solutions it seems as though they are moving along vertices which I think is the same as the $$4 x 4$$ checkerboard remaining on the checkered squares.

Something I'm not sure about is whether king movements might be more apt description than ants because I'm not convinced that $$52$$ possibilities exist for the aforementioned case without throwing a wrench into the works somewhere.

• Have you fully taken into account condition "without intersecting opposite paths at their middle points." which is much less restrictive than "never intersecting" ? Commented Jul 22 at 4:35

There is one case where these ants take the first two vertices on the diagonal, which must be paired with the one case where the other (northeast to southwest) ants take the last two. Similarly, one where these ants take the last two, and the other ants take the first two. So far we have contributed $$1^2 + 1^2 = 2$$ to our count.
There are four cases where these ants take vertices 1 and 3 on the diagonal, paired with four cases where the other ants take vertices 2 and 4. That contributes $$4^2$$. Similarly, four cases where these ants take 2 and 4 and the other ants take 1 and 3, for another $$4^2$$.
There is one case where these ants take vertices 1 and 4 on the diagonal, paired with $$9$$ where the other ants take 2 and 3, similarly $$9$$ where these ants take 2 and 3 paired with $$1$$ where the others take 1 and 4. That makes a total of $$1^2 + 1^2 + 4^2 + 4^2 + 9 + 9 = 52$$, which is $$a(3)$$ in the sequence.