UVa 900

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 $n\leq 50$ 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

Solution

Solution link