Fibonacci Numbers


The Fibonacci numbers Fn are defined recursively by the relation Fn = Fn–1 + Fn–2, where F1 = F2 = 1. The first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …

They describe, among other things, the number of ways to tile a 2 x (n–1) checkerboard with 2 x 1 dominoes.

There’s only one way to tile a 2 x 1 board:

fibboard1

 

 

 

 

Two ways to tile a 2 x 2 board:

fibboard2

 

 

 

 

Three ways to tile a 2 x 3 board:

fibboard3

 

 

 

 

Five ways to tile a 2 x 4 board:

fibboard4

 

 

 

 

 

Eight ways to tile a 2 x 5 board:

fibboard5

 

 

 

 

 

 

Thirteen ways to tile a 2 x 6 board:

fibboard6

 

 

 

 

 

 

 

 

Twenty-one ways to tile a 2 x 7 board:

fibboard7

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