1. a0=0
a1=2 an
a2=6
an-1
a3=12
a4=20 a2 a1 a0

2. 10.1 The First-Order Linear Recurrence Relation
geometric progression
a first-order, linear, and homogeneous recurrence relation
(difference equation) with constant coefficient
The arithmetic progression
is nonhomogeneous.

3. 10.1 The First-Order Linear Recurrence Relation
There are many sequences that satisfy
For example, 5,15,45,135,... or 7,21,63,189,.... To pinpoint
the particular sequence described, we need to know one of the
terms of the sequence. (boundary condition, or initial condition
since usually a0 is specified)
determines the sequence 5,15,45,135,...

4. 10.1 The First-Order Linear Recurrence Relation
Ex A bank pays 6% (annual) interest on savings, compounding
the interest monthly. If Boonie deposits $1000 on the first day of
May, how much will this deposit be worth a year later?
Let pn denote the value of deposit at the end of n months.
Then pn+1=pn+(6%/12)pn=1.005pn. With p0=$1000, we have
pn=p0(1.005)n. The answer is p12=$1000(1.005)12=$
A nonlinear recurrence relation

5. 10.1 The First-Order Linear Recurrence Relation
nonhomogeneous linear recurrence relation
Ex time complexity of bubble sort algorithm
an=an-1+(n-1), n>1, a1=0, where
an=the number of comparisons to sort n numbers
an- an-1= n-1
an-1- an-2= n-2
an-2- an-3= n-3
a2- a1= 1 +
an = (n-1)=(n2-n)/2

6. 10.1 The First-Order Linear Recurrence Relation
find the recurrence pattern
nonconstant coefficients
Ex. 10.5
Ex. 10.6
a0=0
a1- a0=2
a1=2
a2 -a1=4
a2=6
a3- a2=6
a3=12
a4=20
a4- a3=8
an- an-1=2n +
an=n2+n

7. 10.2 The Second-Order Linear Homogeneous Recurrence Relation
with Constant Coefficients
three cases for the characteristic roots:
(a) distinct real numbers
(b) complex conjugates
(c) duplicate roots

8. Case (A): distinct real roots

9. Case (A): distinct real roots

10. Case (A): distinct real roots
Ex number of legal arithmetic expressions, without
parentheses, that are made up of the digits 0,1,2,...,9 and the binary
operator +,*,/.
Let an be the number of expressions made up of n symbols. Then
a1=10 (0,1,...,9), a2=100 (00,01,...,99). For n>2, two cases:
(a) the last two symbols are digits: remove the last digit, we have a
legal expression for an-1. (10an-1)
(b) the last two symbols are operator and digit: remove the two
symbols, we have a legal expression for an-2 (29an-2, no /0)
Therefore, an=10an-1+29an-2, and a1=10, a2=100.

11. Case (A): distinct real roots
Ex Find a recurrence relation for the number of binary
sequences of length n that have no consecutive 0's.
Let an be the number of such sequences with length n,n>0.
Then a1=2, a2=3. There are two cases for an:
(1) the nth symbol is 1: the preceding n-1 symbols sequence is
counted in an-1
(2) the nth symbol is 0: an ends in 10 and the preceding n-2 symbols
sequence is counted in an-2.
Therefore, an=an-1+an-2.

12. 10.2 The Second-Order Linear Homogeneous Recurrence Relation
with Constant Coefficients
Be careful not to draw conclusions from a few (or even,
perhaps, many) particular instances.
Ex Arrange pennies contiguously in each row where each
penny above the bottom row touches two pennies in the row
below it.
a1=1,a2=1,a3=2,a4=3,a5=5,a6=8,... Is an=Fn? NO

13. 10.2 The Second-Order Linear Homogeneous Recurrence Relation
with Constant Coefficients
extend to higher order

14. 10.2 The Second-Order Linear Homogeneous Recurrence Relation
with Constant Coefficients
Case (b) Complex Roots
DeMoivre's Theorem y x

15. Case (b) Complex Roots

16. Case (b) Complex Roots
=bDn-1-b2Dn-2

17. Case (c) Repeated Real Roots

18. 10.3 The Nonhomogeneous Recurrence Relation

19. 10.3 The Nonhomogeneous Recurrence Relation

20. 10.3 The Nonhomogeneous Recurrence Relation
peg peg peg 3
n+1 disks
move to
rule: one disk at a time
larger one must not on top of a smaller one

21. 10.3 The Nonhomogeneous Recurrence Relation
Ex Pauline takes out a loan of S dollars that is to be paid
back in T periods of time. If i is the interest rate per period for the
loan, what constant payment P must she make at the end of each
period?
an:the amount still owed on the loan at the end of the nth
period (following the nth payment)
Example:
S=5,000,000
i=10%/12
T=20 years=240 months
P=48251

22. 10.3 The Nonhomogeneous Recurrence Relation
Ex The snowflake curve 1 1 1
area:

23. 10.3 The Nonhomogeneous Recurrence Relation
Ex The snowflake curve

24. Summary
r=1
1. Linear combinations work.
2. If f(n) contains rn and r is a characteristic root of multiplicity k,
then multiply by nk.

25. 10.3 The Nonhomogeneous Recurrence Relation
Ex For n>1 suppose that there are n people at a party and
that each of these people shakes hands (exactly one time) with all
of the other people there (and no one shakes hands with himself
or herself). If an counts the total number of handshakes, then
an+1=an+n, a2=1, n>1, because when the (n+1)th person arrives,
he or she will shake hands with the n other people already arrived.

26. 10.3 The Nonhomogeneous Recurrence Relation

27. 10.4 The Method of Generating Functions

28. 10.4 The Method of Generating Functions

29. 10.4 The Method of Generating Functions

30. 10.5 A Special Kind of Nonlinear Recurrence Relation

31. 10.5 A Special Kind of Nonlinear Recurrence Relation
Ex (continued)
for n+1 vertices:bn+1
select one as root
b bn
b bn-1
bn b0

32. 10.5 A Special Kind of Nonlinear Recurrence Relation
Ex (continued)

33. 10.5 A Special Kind of Nonlinear Recurrence Relation
Ex (continued)

34. 10.5 A Special Kind of Nonlinear Recurrence Relation
Ex Use stacks to permute the ordered list 1,2,...,n.
output
1,2,...,n input
We can generate 1,2 or 2,1 from 1,2.
We can not generate 3,1,2 from 1,2,3.
stack
Let an count the number of ways to
permute 1,2,...,n using this method.

35. 10.5 A Special Kind of Nonlinear Recurrence Relation
Ex Use stacks to permute the ordered list 1,2,...,n.
Suppose the output list for 1,2,...,n,n+1is: 1
j numbers
k numbers
j+k=n n
n-1 n

36. Exercise:P423:10
P432,433:1,6
P444: 12
P457: 6