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December 2010

PrefixSet edition (2010-10)

Count tilings of a narrow but long rectangle with tiles of size 1x1 or 2x2.

A board consisting of squares arranged into N rows and M columns is given. A *tiling* of this board is a pattern of tiles that covers it. A tiling is *interesting* if:

- only tiles of size 1x1 and/or 2x2 are used;
- each tile of size 1x1 covers exactly one whole square;
- each tile of size 2x2 covers exactly four whole squares;
- each square of the board is covered by exactly one tile.

For example, the following images show a few interesting tilings of a board of size 4 rows and 3 columns:

Two interesting tilings of a board are *different* if there exists at least one square on the board that is covered with a tile of size 1x1 in one tiling and with a tile of size 2x2 in the other. For example, all tilings shown in the images above are different.

Write a function:

class Solution { public int solution(int N, int M); }

that, given two integers N and M, returns the remainder modulo 10,000,007 of the number of different interesting tilings of a board of size N rows and M columns.

For example, given N = 4 and M = 3, the function should return 11, because there are 11 different interesting tilings of a board of size 4 rows and 3 columns:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [1..1,000,000];
- M is an integer within the range [1..7].

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