Fibonacci Numbers

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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:
|

Two ways to tile a 2 x 2 board:
=, ||

Three ways to tile a 2 x 3 board:
|||, |=, =|

Five ways to tile a 2 x 4 board:
||||, ||=, |=|, =||, ==

Eight ways to tile a 2 x 5 board:
|||||, |||=, ||=|, |=||, =|||, |==, =|=, ==|

Thirteen ways to tile a 2 x 6 board:
Thirteen 2-by-6 boards

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

See also Graham, Knuth, and Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

For a practical, commercial use of Fibonacci numbers to encode binary data in a self-clocking code, visit Kronos, Incorporated.

Designed and rendered long ago using Mathematica 3.0 and GifOMatic for NeXT.
Copyright © 1997–2002 Robert M. Dickau.

(With belated thanks to Steven Fairgrieve for some minor corrections.)

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