Recursive Sequences Lecture 24 Section 5.6 Robb T. Koether Hampden-Sydney College Wed, Feb 27, 2013 Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 1 / 21
1 Recursive Sequences The Fibonacci Sequence The Catalan Numbers The Towers of Hanoi 2 Assignment Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 2 / 21
Outline 1 Recursive Sequences The Fibonacci Sequence The Catalan Numbers The Towers of Hanoi 2 Assignment Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 3 / 21
Recursive Sequences Definition (Recursive Sequence) A sequence a 0, a 1, a 2,... is recursive if, for all i k for some integer k > 0, each term a i is defined in terms of certain terms a j with j < i. The initial conditions specify the values of a 0, a 1,..., a k 1. In most examples, we specify the first one or two terms and then define all subsequent terms in terms of the previous one or two terms. Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 4 / 21
Outline 1 Recursive Sequences The Fibonacci Sequence The Catalan Numbers The Towers of Hanoi 2 Assignment Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 5 / 21
The Fibonacci Sequence The Fibonacci sequence is a well known example. Define F 0 = 0 and F 1 = 1. Define recursively F n = F n 1 + F n 2 for all n 2. The first few terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 6 / 21
Outline 1 Recursive Sequences The Fibonacci Sequence The Catalan Numbers The Towers of Hanoi 2 Assignment Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 7 / 21
The Catalan Numbers Another example is the Catalan numbers. Define C 0 = 1. Define recursively C n = 2 ( 2n 1 n+1 The first few terms are ) C n 1 for all n 1. 1, 1, 2, 5, 14, 42, 132. Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 8 / 21
The Catalan Numbers The Catalan numbers can also be defined nonrecursively as C n = 1 ( ) 2n. n + 1 n They can also be defined recursively as C 0 = 1 and for all n 1. C n = C 0 C n 1 + C 1 C n 2 + + C n 1 C 0, Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 9 / 21
The Catalan Numbers How many distinct binary trees are there with exactly n vertices. Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 10 / 21
Binary trees with 1 vertex Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 11 / 21
Binary trees with 2 vertices Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 11 / 21
Binary trees with 3 vertices Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 11 / 21
Binary trees with 4 vertices Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 11 / 21
B A How many paths are there from A to B, moving north and east? Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
B A Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
B A Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
B A Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
B A Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
B A Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
B A Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
B A Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
B A Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
B A Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
B A Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
B A Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
B A Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
B A Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
B A Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 12 / 21
Outline 1 Recursive Sequences The Fibonacci Sequence The Catalan Numbers The Towers of Hanoi 2 Assignment Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 13 / 21
The Towers of Hanoi puzzle has three pegs (Peg 1, Peg 2, Peg 3) with n disks stacked on Peg 1. A legal move is to move a single disk to another peg without placing a larger disk on a smaller disk. The goal is to assemble all n disks on Peg 3. How many moves will it take? Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 14 / 21
Peg 1 Peg 2 Peg 3 All 8 disks on Peg 1 Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 15 / 21
Peg 1 Peg 2 Peg 3 Reassemble 7 disks on Peg 2 Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 15 / 21
Peg 1 Peg 2 Peg 3 Move largest disk to Peg 3 Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 15 / 21
Peg 1 Peg 2 Peg 3 Reassemble 7 disks on Peg 3 Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 15 / 21
We see that we can solve the puzzle for 8 disks if we can solve it for 7 disks. So we can solve it for 7 disks if we can solve it for 6 disks. And so on, down to 2 disks: we can solve it for 2 disks if we can solve it for 1 disk. How do we solve it for 1 disk? Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 16 / 21
Peg 1 Peg 2 Peg 3 The solution for 8 disks using the solution for 6 disks Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 17 / 21
Peg 1 Peg 2 Peg 3 Reassemble 6 disks on Peg 3 Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 17 / 21
Peg 1 Peg 2 Peg 3 Move Disk 7 to Peg 2 Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 17 / 21
Peg 1 Peg 2 Peg 3 Reassemble 6 disks on Peg 2 Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 17 / 21
Peg 1 Peg 2 Peg 3 Move Disk 8 to Peg 3 Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 17 / 21
Peg 1 Peg 2 Peg 3 Reassemble 6 disks on Peg 1 Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 17 / 21
Peg 1 Peg 2 Peg 3 Move Disk 7 to Peg 3 Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 17 / 21
Peg 1 Peg 2 Peg 3 Reassemble 6 disks on Peg 3 Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 17 / 21
How many moves are required to solve the puzzle for n disks? Let m n be the number of moves required when there are n disks. When n = 1: m 1 = 1. When n = 2: m 1 + 1 + m 1 = 3. When n = 3: m 2 + 1 + m 2 = 7. In general, m n = 2m n 1 + 1. Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 18 / 21
We can show by induction that m n = 2 n 1. Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 19 / 21
Outline 1 Recursive Sequences The Fibonacci Sequence The Catalan Numbers The Towers of Hanoi 2 Assignment Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 20 / 21
Assignment Assignment Read Section 5.6, pages 290-301. Exercises 2, 5, 11, 14, 22, 23, 27, 28, 33, page 302. Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 21 / 21
Collected Homework 6 Collected Homework 6 Exercises 28, 44, page 242. Exercises 7, 14, page 256. Exercises 18, 34, page 266. Exercises 6, 8, page 277. Robb T. Koether (Hampden-Sydney College) Recursive Sequences Wed, Feb 27, 2013 22 / 21