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Centaurs on a chess board

Published on Sunday, 3rd April 2016, 01:00 am; Solved by 172;
Difficulty rating: 80%

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.

p554-centaurs.png

It can be shown that at most n2 non-attacking centaurs can be placed on a board of size 2n×2n.
Let C(n) be the number of ways to place n2 centaurs on a 2n×2n board so that no centaur attacks another directly.
For example C(1) = 4, C(2) = 25, C(10) = 1477721.

Let Fi be the ith Fibonacci number defined as F1 = F2 = 1 and Fi = Fi-1 + Fi-2 for i > 2.

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