John Baez mused that my formula might be right after all! But it turns out that it can hardly be correct. As Christopher Upshaw noticed, the whole Ansatz is pretty weird. Maybe even weird enough to warrant further inspection for its own sake... Nevertheless,

I gave a this formula:

\$n = \sum_i^k a_i \cdot b_i \$

Which supeficially looks like it could be a special case, coming from another formula, for matrix multiplication (matrix powers, so relevant):

\$c_{ij} = \sum_k^n a_{ik}\cdot b_{kj} \$

And while I mentioned upthread how my formula undercounts, there's another problem, a weakness John had spotted right away:

> 1. how you are reducing the case of a general graph to the simpler sort of graph you're discussing

I had proposed to 'simply' look at all possible loops in the graph. But that is an infinite set! Put like this it is actually the set of all paths of the kind we're looking for!

That's not what I had in mind, I thought that there'd be a finite number of loops until the whole graph would be covered. But, as my mentioned realization suggests, finding all paths _is the problem_!

Jeez #-D :_S (:-) Cheers!