A399700

[John Mason]

Dear all,
Consider a wheel constructed from a circle of diameter 2n+1 units, surrounded by a rim. Cross the circle with horizontal bars, each of thickness 1 unit, and each at 1 unit from the next. Superimpose another identical wheel. The sequence A399700 maps the variation over time of the number of resulting holes as the second wheel rotates. See an example below.

Pontus von Brömssen pointed out that the number of variations of the number of holes as the second wheel rotates between 0 and 90 degrees is floor (((n + 1)^2) / 4) + 1. I found this very surprising and I have no idea how to explain it.

The formula even works when, for example, the number of bars is 7 and the number of holes sometimes decreases: 8 10 14 18 22 26 30 34 36 40 44 48 46 50 52 56 52

I have checked the formula through to bars = 200.

Any ideas?

john