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Problem 743

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Window into a Matrix

A window into a matrix is a contiguous sub matrix.

Consider a $2\times n$ matrix where every entry is either $0$ or $1$.
Let $A(k,n)$ be the total number of these matrices such that the sum of the entries in every $2\times k$ window is $k$.

You are given that $A(3,9) = 560$ and $A(4,20) = 1060870$.

Find $A(10^8,10^{16})$. Give your answer modulo $1\ 000\ 000\ 007$.

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矩阵窗口

矩阵的窗口是指矩阵中一个连续的子矩阵。

考虑一个$2\times n$的矩阵，其中的元素均为$0$或$1$。
若矩阵中任意一个$2\times k$的窗口中元素之和都为$k$，记所有这样的矩阵数目为$A(k,n)$。

已知$A(3,9) = 560$，$A(4,20) = 1060870$。

求$A(10^8,10^{16})$，并将你的答案对$1\ 000\ 000\ 007$取余。

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