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RecurrenceRecurrence

Relations

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Roxanne Scott

Iveta Ovsepyan

Deborah Ellis

Roxanne Scott

Iveta Ovsepyan

Deborah Ellis

04/19/23

A DefinitionA DefinitionA DefinitionA Definition

A recurrence relation is a way of defining a sequence of numbers by defining the first term(s) and then prescribing an algorithm that shows us how to compute an in terms of ai for i < n.

A recurrence relation is a way of defining a sequence of numbers by defining the first term(s) and then prescribing an algorithm that shows us how to compute an in terms of ai for i < n.

FibonacciLet f1 = 1, f2 = 1, and for all

n > 2

Let fn = f n-2 + fn-1

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LucasLucas

Let L1 = 1, L2 = 3

And for all

n > 2

Let Ln = L n-2 + L n-1

Let L1 = 1, L2 = 3

And for all

n > 2

Let Ln = L n-2 + L n-1

Let P0 = 0, P1=1

and let

Pn = 2Pn-1 + Pn-2

Let P0 = 0, P1=1

and let

Pn = 2Pn-1 + Pn-2

Pell

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= # { U \in {1,2,3,….,n} : |U| = k

Consider:

= # { U \in {1,2,3,….,n} : |U| = k

Consider:

nk “ n choose k ”

32 =

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n

0 XX XX XX XX XX XX

1 XX XX XX XX XX

2 XX XX XX XX

3 XX XX XX

4 XX XX

5 XX

6

k 0 1 2 3 4 5 6

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Recurrence Relations

Recurrence Relations

Recurrence Relations

cuRelations

Re r encer