## Counting block combinations II

### Problem 115

Published on Friday, 24th February 2006, 06:00 pm; Solved by 6158; Difficulty rating: 35%NOTE: This is a more difficult version of Problem 114.

A row measuring *n* units in length has red blocks with a minimum length of *m* units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one black square.

Let the fill-count function, F(*m*, *n*), represent the number of ways that a row can be filled.

For example, F(3, 29) = 673135 and F(3, 30) = 1089155.

That is, for *m* = 3, it can be seen that *n* = 30 is the smallest value for which the fill-count function first exceeds one million.

In the same way, for *m* = 10, it can be verified that F(10, 56) = 880711 and F(10, 57) = 1148904, so *n* = 57 is the least value for which the fill-count function first exceeds one million.

For *m* = 50, find the least value of *n* for which the fill-count function first exceeds one million.