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Problem 670

Colouring a Strip

A certain type of tile comes in three different sizes - $1\times 1$, $1\times2$, and $1\times 3$ - and in four different colours: blue, green, red and yellow. There is an unlimited number of tiles available in each combination of size and colour.

These are used to tile a $2\times n$ rectangle, where $n$ is a positive integer, subject to the following conditions:

• The rectangle must be fully covered by non-overlapping tiles.
• It is not permitted for four tiles to have their corners meeting at a single point.
• Adjacent tiles must be of different colours.

For example, the following is an acceptable tiling of a $2\times 12$ rectangle:

but the following is not an acceptable tiling, because it violates the “no four corners meeting at a point” rule:

Let $F(n)$ be the number of ways the $2\times n$ rectangle can be tiled subject to these rules. Where reflecting horizontally or vertically would give a different tiling, these tilings are to be counted separately.

For example, $F(2) = 120$, $F(5) = 45876$, and $F(100)\equiv 53275818 \pmod{1,000,004,321}$.

Find $F(10^{16}) \bmod 1,000,004,321$.

彩砖铺盖

• 长方形必须完全被覆盖，且砖块之间不能重叠。
• 允许有四块砖块共用一个顶点。
• 相邻的砖块必须有不同的颜色。