979. Heptagon Hopping

The hyperbolic plane, represented by the open unit disc, can be tiled by heptagons. Every tile is a hyperbolic heptagon (i.e. it has seven edges which are segments of geodesics in the hyperbolic plane) and every vertex is shared by three tiles.
Please refer to Problem 972 for some of the definitions.

The diagram below shows an illustration of this tiling.

Now, a hyperbolic frog starts from one of the heptagons, as shown in the diagram. At each step, it can jump to any one of the seven adjacent tiles.

Define $F(n)$$F(n)$ to be the number of paths the frog can trace so that after $n$$n$ steps it lands back at the starting tile.
You are given $F(4)=119$$F(4) = 119$.

Find $F(20)$$F(20)$.

979. 七边形跳跃

双曲平面（此处用 开单位圆盘 表示）可以用七边形地砖密铺——每块地砖都是双曲七边形（换句话说，每个地砖的七条边都是这双曲平面的一条 测地线 的一部分）且每个交点均被三块地砖共享。
如需某些定义的进一步阐释，请参考 972 题。

下图展示了题述的密铺方法。

现在，一只双曲青蛙从图中所示的地砖出发。在接下来的每步中，这只青蛙可以跳向七个相邻地砖中的任意一块。

定义 $F(n)$$F(n)$ 为：青蛙可走的路线数，使得经过 $n$$n$ 步后，它能回到起始位置。已知 $F(4)=119$$F(4) = 119$。

求 $F(20)$$F(20)$。

点 这个链接 回到源站。

点 这个链接 回到详细版题目目录。