Amoebas in a 2D grid (noch nicht ├╝bersetzt)

Problem 762

Consider a two dimensional grid of squares. The grid has 4 rows but infinitely many columns.

An amoeba in square $(x, y)$ can divide itself into two amoebas to occupy the squares $(x+1,y)$ and $(x+1,(y+1) \bmod 4)$, provided these squares are empty.

The following diagrams show two cases of an amoeba placed in square A of each grid. When it divides, it is replaced with two amoebas, one at each of the squares marked with B:

Originally there is only one amoeba in the square $(0, 0)$. After $N$ divisions there will be $N+1$ amoebas arranged in the grid. An arrangement may be reached in several different ways but it is only counted once. Let $C(N)$ be the number of different possible arrangements after $N$ divisions.

For example, $C(2) = 2$, $C(10) = 1301$, $C(20)=5895236$ and the last nine digits of $C(100)$ are $125923036$.

Find $C(100\,000)$, enter the last nine digits as your answer.