## Centaurs on a chess board

### Problem 554

On a chess board, a centaur moves like a king or a knight. The diagram below shows the valid moves of a centaur (represented by an inverted king) on an 8x8 board.

It can be shown that at most `n`^{2} non-attacking centaurs can be placed on a board of size 2`n`×2`n`.

Let `C`(`n`) be the number of ways to place `n`^{2} centaurs on a 2`n`×2`n` board so that no centaur attacks another directly.

For example `C`(1) = 4, `C`(2) = 25, `C`(10) = 1477721.

Let `F _{i}` be the

`i`

^{th}Fibonacci number defined as

`F`

_{1}=

`F`

_{2}= 1 and

`F`

_{i}=

`F`

_{i-1}+

`F`

_{i-2}for

`i`> 2.

Find $\displaystyle \left( \sum_{i=2}^{90} C(F_i) \right) \text{mod } (10^8+7)$.