Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Fibonacci Example: A young pair of rabbits (1 male, 1 female) arrive on a
deserted island. They can breed after they are two months old and produce
another pair. Thereafter each pair at least two months old can breed once a
month. How many pairs f n of rabbits are there after n months.
• n = 1: f 1 = 1 Initial
• n = 2: f 2 = 1
conditions
• n > 2: f n = f n-1 + f n-2 Recurrence relation
The n > 2 formula is true since each new pair comes from a pair at least 2
months old.
Example: For bit strings of length n < 3, find the recurrence relation and initial
conditions for the number of bit strings that do not have two consecutive 0’s.
• n = 1: a 1 = 2 Initial {0,1}
• n = 2: a 2 = 3 conditions {01,10,11}
• n > 2: a n = a n-1 + a n-2 Recurrence relation
For n > 2, there are two cases: strings ending in 1 (thus, examine the n41 case)
and strings ending in 10 (thus, examine the n42 case).
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Definition: A linear homogeneous recurrence relation of degree k with constant
coefficients is a recurrence relation of the form
where {c i },R.
a n = c 1 a n41 + c 2 a n42 + … + c k a n4k ,
Motivation for study: This type of recurrence relation occurs often and can be
systematically solved. Slightly more general ones can be, too. The solution
methods are related to solving certain classes of ordinary differential equations.
Notes:
• Linear because the right hand side is a sum of previous terms.
• Homogeneous because no terms occur that are not multiples of a j ’s.
• Constant because no coefficient is a function.
• Degree k because a n is defined in terms of the previous k sequential terms.
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