# Lecture 3: Sequences, Recurrences and Limits

### Background Material

• Fibonacci Numbers
• Recurrence Relations
• Full binary trees - all internal nodes have two children (number of internal nodes is one less than the number of leaf nodes)
• Limits
• Asymptotic Computing Time Analysis

• Maple Help - rsolve, time,

### Topics

• Programming in Maple.
• Maple procedures
• Recursion and "option remember"
• Timing maple commands
• Fibonacci numbers and dominoes - how many ways are there to fill a 2 X n rectangle with 1 X 2 dominoes (placed vertically or two stacked horizontally)?
• The golden ratio and solving the Fibonacci recurrence
• Empirical investigation of asymptotic computing times.
• Computing time of two algorithms to compute the Fibonacci numbers

### Maple worksheets and documentation

• fib.mw - Maple worksheet investigating the Fibonacci numbers. Also contains a recursive function to generate all configurations of dominoes that fill a 2 X n rectangle.

### Assignments

1. assign1.mw - first assignment to be handed in.
Created: Oct. 20, 2005 by jjohnson@cs.drexel.edu