John wrote:
> \[ 1, 0, 2, 0, 4, 0, 8, 0, 16, 0, \dots \]

Aha -- [rose graphs](https://en.wikipedia.org/wiki/Rose_(topology))! (Except each petal has its own vertex, rather than all petals being self-loops.)

**EDIT**: You added more zeroes when I refreshed. :(