Jim Lambers
Math 6A
Spring Quarter 2003-04
Lecture 22 Examples
This example corresponds to Section 6.1 in the text.
Example (Exercise 23(a), Section 6.1) Find a recurrence relation that describes the number of bit
strings of length nthat have two consecutive zeros.
Solution Let anbe the number of bit strings of length nthat have two consecutive zeros, where n
is a nonnegative integer. To find a recurrence relation that is satisfied by {an}, we must determine
how ancan be computed if we know anโ1,anโ2and so on. This is equivalent to determining how
bit strings of length n, that have two consecutive zeros, can be obtained from strings of shorter
length.
A bit string of length nโ1 is extended to a bit string of length nby adding a 0 or a 1.
Furthermore, such a bit string of length nโ1 may, or may not, have two consecutive zeros. It
follows that all bit strings of length nfall into exactly one of four categories:
1. Strings that have two consecutive zeros in their first nโ1 bits, and end with a 0
2. Strings that have two consecutive zeros in their first nโ1 bits, and end with a 1
3. Strings that do not have two consecutive zeros in their first nโ1 bits, and end with a 0
4. Strings that do not have two consecutive zeros in their first nโ1 bits, and end with a 1
Strings in the fourth category cannot have two consecutive zeros, so we do not need to consider
them. The remaining strings can be described using ony two categories:
1. Strings that have two consecutive zeros in their first nโ1 bits, and end with a 1
2. All strings that end with a 0
The first category contains anโ1strings, because they have two consecutive zeros among their first
nโ1 bits. Therefore, this category contributes anโ1toward an.
The second category must be examined further. We can decompose it into four categories, as
we did before:
1. Strings that have two consecutive zeros in their first nโ2 bits, and end with a 00
2. Strings that have two consecutive zeros in their first nโ2 bits, and end with a 10
3. Strings that do not have two consecutive zeros in their first nโ2 bits, and end with a 00
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