math.CO/0409509 number 86

### What is the conjecture
The number of unlabeled alternating octopi with $n$ black nodes and $k$ white nodes has the g.f.

$$
\sum_{k,n\ge1} \frac{\phi(k)}{k}\log(\frac{(1-x^ny^n)^2}{1-x^ny^n(3+x^n+y^n)}).
$$

The conjecture is now that the number of those octopi with $n$ black nodes and $n$ white nodes (the diagonal of the above array) is

$$
\begin{equation}-2+3\sum_{d|n}\frac{\phi(n/d)\binom{2d}{d}}{2n}.\end{equation}
$$

See https://oeis.org/A091468

### Prerequisites needed

Status: open

### Choose either option
- [ ] I plan on adding this conjecture to the repository
- [X] This issue is up for grabs: I would like to see this conjecture added by somebody else

Start by inspecting existing formalized conjectures in the repository to find the expected file location, naming, and statement conventions. Translate the displayed generating-function and diagonal formula into Lean, then run the relevant project checks; done means the conjecture is added in the repository's established style and the checks pass.

Written by the indexing model from the issue text.