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Problem 670

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Colouring a Strip

A certain type of tile comes in three different sizes - $1\times 1$, $1\times 2$, 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 $F(100)\equiv 53275818(\mathrm{mod}1000004321)$.

Find $F({10}^{16})mod1000004321$.

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彩砖条形铺盖

某种砖块有三种不同的大小，分别是$1\times 1$、$1\times 2$和$1\times 3$，同时又有四种不同的颜色：蓝色、绿色、红色和黄色。每种大小和颜色组合的砖块都不限量供应。

这些砖块被用于铺盖大小为$2\times n$的长方形，其中$n$为正整数。铺盖过程必须满足以下条件：

• 长方形必须完全被覆盖，且砖块之间不能重叠。
• 不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。

例如，下图是对大小为$2\times 12$的长方形的一种合理铺盖：

而下图则是一种不合理铺盖，因为这种铺盖违背了“四块砖块不允许共用一个顶点”的规则：

记$F(n)$为大小为$2\times n$的长方形上的合理铺盖总数。上下或左右镜像对称的铺盖视为不同的铺盖。

已知，$F(2)=120$，$F(5)=45876$，$F(100)\equiv 53275818(\mathrm{mod}1000004321)$。

求$F({10}^{16})mod1000004321$。

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