UVa 900 - Brick Wall Patterns

900 - Brick Wall Patterns
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Summary
Given wall bricks of size $$2x1$$ units, what is the number of all possible brick patterns for a wall of height $$2$$ units and a length of $$n \le 50$$ units?

Explanation
Let $$f_n$$ be the number of ways to assembly a wall of length $$n$$.

There are two ways of assembling a wall of length $$n$$, and a height of 2 units. We either place a vertical bar at its beginning and assemble a wall of length $$n-1$$ or we place two horizontal bars on top of one another and assemble then a wall of length $$n-2$$. That is,


 * $$f_n := f_{n-1} + f_{n-2}$$.

The base cases are already given in the example, which leads us to:


 * $$f_n:= \begin{cases}

1, &n=1 \\ 2, &n=2 \\ f_{n-1} + f_{n-2}, & n > 2 \\ \end{cases}$$

Which is the fibonacci series, with a slightly different base! (This proof was originally written by Schultz, on 21:20, 26 May 2007 (EDT))

You can solve the series blazingly fast in $$O(n)$$ using a bottom-up dynamic programming solution.

Gotchas
Use a 64-bit field to hold the results; $$f_{50} > 2^{32} - 1$$.

Input
1 2 3 4 5 6 7 50 0

Output
1 2 3 5 8 13 21 20365011074