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Problem 405

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A rectangular tiling

We wish to tile a rectangle whose length is twice its width.
Let T(0) be the tiling consisting of a single rectangle.
For n > 0, let T(n) be obtained from T(n-1) by replacing all tiles in the following manner:

The following animation demonstrates the tilings T(n) for n from 0 to 5:

Let f(n) be the number of points where four tiles meet in T(n).
For example, f(1) = 0, f(4) = 82 and f(10⁹) mod 17⁷ = 126897180.

Find f(10^k) for k = 10¹⁸, give your answer modulo 17⁷.

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给长方形铺地砖

我们希望给一个长是宽的两倍的长方形铺上地砖。
T(0)表示只用一个长方形的铺法。
对于n > 0，T(n)表示以T(n-1)为基础，将后者的每一块地砖作如下替换得到的新铺法：

以下动画演示了n从0到5时的T(n)：

记f(n)是T(n)中有四块地砖相遇的点的数目。
已知f(1) = 0，f(4) = 82，以及f(10⁹) mod 17⁷ = 126897180。

当k = 10¹⁸时，求f(10^k)，并将其模17⁷取余作为你的答案。

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