Recurrence Relations Solving Linear Recurrence Relations€¦ · Recurrence Relations Dr Patrick...
Transcript of Recurrence Relations Solving Linear Recurrence Relations€¦ · Recurrence Relations Dr Patrick...
4.1
Recurrence Relations
Dr Patrick ChanSchool of Computer Science and Engineering
South China University of Technology
Discrete Mathematic
Chapter 4: Advanced Counting Techniques
4.2
Solving Linear Recurrence Relations4.4
Generating Functions
2Ch. 4.1, 4.2 & 4.5
Agenda
� Recurrence Relations
� Modeling with Recurrence Relations
� Linear Nonhomogeneous Recurrence Relations with Constant Coefficients
� Generating Functions
� Useful Facts About Power Series
� Extended Binomial Coefficient
� Extended Binomial Theorem
� Counting Problems and Generating Functions
� Using Generating Functions to Solve Recurrence Relations
3Ch. 4.1, 4.2 & 4.5
Recursion
How many i-products the
families of students in our
school have?
0 0 1 1 1 1 1 0 1 0 00 1 0
2 4 1
16
7How many i-products?
How many
i-pr oduc ts?How many i-pr oduc ts?
How m any
i -produc t s?
How m any
i- produc t s?
How m any
i -produc t s?
4Ch. 4.1, 4.2 & 4.5
Recurrence Relations
� A recurrence relation for a sequence {an} is an equation that expresses an in terms of one or more previous elements (a0, …, an−1)
� A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation
5Ch. 4.1, 4.2 & 4.5
Recurrence Relations
Example 1
� Let {an} be a sequence that satisfies the recurrence relation an = an-1 - an-2 for n = 2, 3, 4, . . . , and suppose that a0 = 3 and a1 = 5. What are a2 and a3?
� From the recurrence relation:
� a2
� a3
= a1 – a0 = 5 - 3 = 2
= a2 – a1 = 2 - 5 = -3
6Ch. 4.1, 4.2 & 4.5
Recurrence Relations
Example 2
� Consider the recurrence relation
an = 2an−1 − an−2, where n ≥ 2
� Which of the following are solutions?� an = 3n
� 2an−1 − an−2 = 2(3(n - 1)) - 3(n - 2) = 3n = an
� an = 2n
� 2an−1 − an−2 = 2(2n-1) – nn-2 = 2n ≠ an
� an = 5
� 2an−1 − an−2 = 2 x 10 – 5 = 5 = an
����
����
����
7Ch. 4.1, 4.2 & 4.5
Recurrence Relations
� The initial conditions for a sequence specify the terms that precede the first term where the recurrence relation takes effect
� For examplean = an-1 + an-2, what is the value of a3?
Answer depends on a0 and a1 (initial conditions)
� a0 = 3 and a1 = 5 :
� a0 = 1 and a1 = 2 :
� A sequence is determined uniquely by
� Recurrence relation
� Initial conditions
a2 = 8, a3 = 13
a2 = 3, a3 = 5
8Ch. 4.1, 4.2 & 4.5
Modeling with Recurrence Relations
Compound Interest
� Growth of saving in a bank account with r% interest per given period
� Sn = Sn-1 + r · Sn-1 = (r+1) · Sn-1
� Example:� Suppose that a person deposits $10,000 in a savings
account at a bank yielding 11 % per year with interest compounded annually. How much will be in the account after 30 years?
� S30 = 1.11S29 = 1.11(1.11S28) = …= (1.11)30 10,000
9Ch. 4.1, 4.2 & 4.5
Modeling with Recurrence Relations
Fibonacci (Rabbits) Numbers � A young pair of rabbits
(one of each sex) is placed on an island
� A pair of rabbits does not breed until they are 2 months old
� After they are 2 months old, each pair of rabbits produces another pair each month
� Pn = Pn−1 + Pn−2
1
1
1.11
1 1.21.1
1 1.3
1.1.11.1
1.2
Younger than two months
At least two
months old
1
2
3
4
5
10Ch. 4.1, 4.2 & 4.5
Modeling with Recurrence Relations
Tower of Hanoi
� Objective
� Get all disks from peg 1 to peg 3
� Rules
� Only move 1 disk at a time
� Never put a larger disk on a smaller one
1 2 3
11Ch. 4.1, 4.2 & 4.5
Modeling with Recurrence Relations
Tower of HanoiMore than 1 steps
More than 1 steps 1 step More than 1 steps
Break Down
Recursive Call Recursive Call
12Ch. 4.1, 4.2 & 4.5
Modeling with Recurrence Relations
Tower of HanoiMore than 1 steps
More than 1 steps 1 step More than 1 steps
Break Down
Recursive Call Recursive Call
13Ch. 4.1, 4.2 & 4.5
Modeling with Recurrence Relations
Tower of HanoiMore than 1 steps
1 step 1 step 1 step
Break Down
Base Case Base Case
14Ch. 4.1, 4.2 & 4.5
Modeling with Recurrence Relations
Tower of Hanoi
The solution
15Ch. 4.1, 4.2 & 4.5
Modeling with Recurrence Relations
Tower of Hanoi
� Let Hn be the number of moves for a stack of n disks.
� Strategy:
� Move top n−1 disks (Hn−1 moves)
� Move bottom disk (1 move)
� Move top n−1 to bottom disk (Hn−1 moves)
� Hn = 2Hn-1 + 1
16Ch. 4.1, 4.2 & 4.5
Modeling with Recurrence Relations
Example � Find a recurrence relation and give initial conditions for the
number of bit strings of length n that do not have two consecutive 0s.
� Let Pn denote the number of bit strings of length n that do not have two consecutive 0s
� Pn = Pn-1 + Pn-2 , n ≥ 3
� P1 = 2, P2 = 3
1
Any bit string of length n - 1 with no two consecutive 0s
10
Any bit string of length n - 2 with no two consecutive 0s
Pn-1
Pn-2
0101 0110
0111 1010
1011 1110
1111
For P4
P3
P21101
17Ch. 4.1, 4.2 & 4.5
Solving Linear Recurrence Relations
� Given Pn = Pn−1 + Pn−2, what is P100?
� It is not easy to calculate
� Need a better solutionwhich is not in
relation form
E.g. Pn = n*10 -1
18Ch. 4.1, 4.2 & 4.5
Solving Linear Recurrence Relations
� Linear Homogeneous Recurrence of Degree k with Constant Coefficients is a recurrence of the form
an = c1an−1 + … + ckan−k
where the ci are all real numbers, and ck ≠ 0
� Linear: the power of all a i term is one
� Homogeneo us: no constant term (no team without ai)
� Recurrence: a sequence {an} which a
nin terms of a
n-1, a
n-2,…
� Degree k: refer to k previous terms an-k
� Constant Coefficients: c1, c2, … independent from n
� The short name is “k-LiHoReCoCo”
∑=
−=k
i
iniac
1
19Ch. 4.1, 4.2 & 4.5
Solving Linear Recurrence Relations
Example
� Mn = Mn−1 + (1.11)Mn−1
� 1-LiHoReCoCo
� Pn = Pn−1 + Pn−2
� 2-LiHoReCoCo
� an = an-5
� 5-LiHoReCoCo
� an =an-1 + (an-2)2
� Not linear
� Hn = 2Hn−1 + 1
� Not homogeneous
� Bn = nBn-1
� Non-constant coefficient
(n is a variable)k-LiHoReCo Co
� Linear: the power of a ll a i term is one
� Homogeneous: no constant term (no team wi thout ai)
� Recurrence: a sequence {an} which an in terms of an-1 , an-2,…
� Degree k: refer to k previous terms an -k
� Constant Coefficients: c1, c2, … independent from n
20Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos
� Given 2-LiHoReCoCo: an = c1an−1 + c2an−2
� Assume an = w1r1n + w2r2
n for n ≥ 0, r1 and r2 are different and some constants w1, w2
21Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos
� Given 2-LiHoReCoCo: an
= c1a
n−1+ c
2a
n−2 , and
a0 = c and a1 = d
� Assume an = w1r1n + w2r2
n for
r1 and r2 are different and some constants w1, w2
an = w1r1n + w2r2
n
c1an-1 = c1(w1r1n-1 + w2r2
n-1)
c2an-2 = c2(w1r1n-2 + w2r2
n-2)
–
–
an - c1an−1 - c2an−2w1 (r1
n - c1r1n-1 - c2r1
n-2)
+ w2 (r2n - c1r2
n-1 - c2r2n-2)
=
w1 (r1n - c1r1
n-1 - c2r1n-2) + w2 (r2
n - c1r2n-1 - c2r2
n-2) = 0
Equal to 0
w1 r1n-2 (r1
2 - c1r1 - c2) + w2 r2n-2 (r2
2 - c1r2 - c2) = 0
22Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos
� Assume w1, w2, r1 and r2 are not 0
� This formula is hold for any n, therefore,
w1 r1n-2 (r1
2 - c1r1 - c2) + w2 r2n-2 (r2
2 - c1r2 - c2) = 0
w1 r1n-2 (r1
2 - c1r1 - c2) + w2 r2n-2 (r2
2 - c1r2 - c2) = 0
w1 r1n-2
w2 r2n-2
(r12 - c1r1 - c2) + (r2
2 - c1r2 - c2) = 0
r12 - c1r1 - c2 = 0 r2
2 - c1r2 - c2 = 0and
an = w1r1n + w2r2
n
23Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos
� If w1 and w2 are both 0,
� an is a special sequence and all values are 0
� If either w1 or w2 is 0, or either r1 or r2 is 0 (as r1 and r2 are different)
an = w1r1n + w2r2
n
w1 r1n-2 (r1
2 - c1r1 - c2) + w2 r2n-2 (r2
2 - c1r2 - c2) = 0
w1 r1n-2 (r1
2 - c1r1 - c2) = 0
w2 r2n-2 (r2
2 - c1r2 - c2) = 0
r12 - c1r1 - c2 = 0
r22 - c1r2 - c2 = 0
w1 or r1 is 0 (w2 and r2 are non-zero)
w2 or r2 is 0 (w1 and r1 are non-zero)
(w1 and r2) or (w2 and r1) is 0
an is a special sequence and all values are 0
24Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos
� Given 2-LiHoReCoCo: an = c1an−1 + c2an−2 , and a0 = c and a1 = d
� Assume an = w1r1n + w2r2
n for r
1and r
2are different and some constants w
1, w
2
� We know that r1
2 - c1r
1- c
2= 0 and r
22 - c
1r
2- c
2= 0
� Characteristic Equation: r2 - c1r - c2 = 0
� Characteristic Roots: r1 and r2
� w1 and w2 can be calculated by using c and d
a0 = c = w1r10 + w2r2
0
a1 = d = w1r11 + w2r2
1
25Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos
Theorem
� Consider an arbitrary 2-LiHoReCoCo:
an = c1an−1 + c2an−2
� By substituting an = rn, we have the characteristic equation:
r2 − c1r − c2 = 0
� If there has two different roots r1 and r2, then an = w1r1
n + w2r2n
for n ≥ 0 and some constants w1, w2
26Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos
� Proof
Given r1, r2 are the characteristic root
where and w1, w2 are constants
�Two steps for the proof
1. Show if an = w1r1n + w2r2
n , {an} is a solution of the recurrence relation
2. Show if {an} is the solution of the recurrence
relation, an = w1r1n + w2r2
n
an = c1an−1 + c2an−2 an = w1r1n + w2r2
n⇔⇔⇔⇔
27Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos
� Step 1Show if an = w1r1
n + w2r2n , {an} is a solution of the
recurrence relation
2211 −− + nnacac ( ) ( )2
22
2
122
1
22
1
111
−−−− +++= nnnnrwrwcrwrwc
( ) ( )221
2
22211
2
11 crcrwcrcrwnn +++= −−
2
2
2
22
2
1
2
11 rrwrrwnn −− +=
nnrwrw 2211 +=
na=
021
2 =−− crcrr1 and r2 are the so lu tion of
28Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos
� Step 2Show if {an} is the solution of the recurrence relation, an = w1r1
n + w2r2n
� Suppose that {an} is a solution of the recurrence relation, and the initial conditions a0 = C0 and a1 = C1 hold
� We want to show that there are constants w1 and w2 such that the sequence {an} with an = w1r1
n + w2r2n satisfies these same initial
conditions
� a0 = C0 = w1 + w2 and a1 = C1 = w1r1 + w2r2
� By solving these two equations:
� When r1 ≠ r2, {an} with w1r1n + w2r2
n satisfy the 2 initial conditions
21
2011
rr
rCCw
−
−=
21
1202
rr
CrCw
−
−=
29Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos
� We know that {an} and {α1r1n + α2r2
n} are both solutionsof the recurrence relation an = c1an−1 + c2an−2 and both
satisfy the initial conditions when n = 0 and n = 1
� Because there is a unique solution of 2-LiHoReCoCo
with two initial conditions, it follows that the two
solutions are the same, that is, an = α1r1n + α2r2
n for all nonnegative integers n
� We have completed the proof
30Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos
Example 1� Solve the recurrence an = an−1 + 2an−2 given the initial
conditions a0 = 2, a1 = 7
� Characteristic Equation: r2 − r − 2 = 0
� Characteristic Root:
� r = (1 ± 3) / 2
� r = 2 or r = −1
� Therefore, an = w1 2n + w2 (−1)n
� By using a0 = 2, a1 = 7
� a0 = 2 = w120 + w2 (−1)0
� a1 = 7 = w121 + w2 (−1)1
� w1 = 3 and w2 = 1
� Therefore, an = 3·2n − (−1)n
a
acbbx
2
42 −±−
=
2-LiHoReCoCo: an = c1an−1 + c2an−2 ,
Characteristic Equation: r2 - c1r - c2 = 0
an = w1r1n + w2r2
n (r1 and r2 are different)
31Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos
Example 2� Find an explicit formula for the Fibonacci numbers
� Recall fn = fn-1 + fn-2
� Characteristic equation: r2 - r - 1 = 0
� Characteristic roots:
� Initial conditions f0 = 0 and f 1 = 1
2/)51(1 +=r 2/)51(2 −=rnn
nwwf
−+
+=
2
51
2
5121
2100 wwf +==
−+
+==
2
51
2
511 211 wwf 5
12
−=w5
11=w
nn
nf
−−
+=
2
51
5
1
2
51
5
1
2-LiHoReCoCo: an = c1an−1 + c2an−2 ,
Characteristic Equation: r2 - c1r - c2 = 0
an = w1r1n + w2r2
n (r1 and r2 are different)
32Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos with two same roots
Theorem
� Let c1 and c2 be real numbers with c2 ≠ 0. Suppose that r2 – c
1r - c
2= 0 has only one
root r0
� A sequence {an} is a solution of the recurrence relation an = c1an-1 + c2an-2 if and only if an = w1r0
n + w2nr0n, for n = 0, 1, 2, ...,
where w1 and w2 are constants
33Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos with two same roots
Example 1� What is the solution of the recurrence relation
an = 6an-1 - 9an-2 with initial conditions a0 = 1 and a1 = 6?
� Characteristic equation: r2 - 6r + 9 = 0
� Only one characteristic root: r = 3
� Hence, the solution to this recurrence relation is
an = w13n + w2n3n
for some constants α1 and α2
� By using the initial conditions,
a0 = 1 = w1, a1 = 6 = 3w1 + 3w2, so w1=1 and w2 = 1
� Consequently, an = 3n + n3n
2-LiHoReCoCo: an = c1an−1 + c2an−2 ,
Characteristic Equation: r2 - c1r - c2 = 0
an = w1r0n + w2 n r0
n
34Ch. 4.1, 4.2 & 4.5
Solving 2-LiHoReCoCos
Summary
� Given : an = c1an−1 + c2an−2 and a0 = c and a1 = d
1. Characteristic equation: r2 − c1r − c2 = 0
2a. If Characteristic Root (r1 and r2) are different
1. an = w1r1n + w2r2
n is the solution
2. Use a0 = w1 + w2 = c and a1 = w1r1 + w2r2 = d to
solve w1 and w2
2b. If Characteristic Root (r 1 and r2) are the same
1. an = w1rn + w2 n rn is the solution
2. Use a0 = w1 = c and a1 = w1r1 + w2r2 = d to solve w1 and w2
35Ch. 4.1, 4.2 & 4.5
☺☺☺☺ Small Exercise ☺☺☺☺
� What is the solution of the recurrence relation a
n= - a
n + 6a
n-2with initial conditions a
0= 0
and a1 = 5?
� What is the solution of the recurrence relation an = - 2an - an-2 with initial conditions a0 = 5 and a1 = -6?
36Ch. 4.1, 4.2 & 4.5
☺☺☺☺ Small Exercise ☺☺☺☺
� the recurrence relation: an = - an + 6an-2
� Initial conditions a0 = 0 and a1 = 5
� Characteristic Equation:
� Characteristic Root:
� Therefore,
� Using the initial condition� a0 = 0 = w1 + w2
� a1 = 5 = -3w1 + 2w2
� w1 = -1, w2 = 1
� Therefore,
(r + 3)(r – 2) = 0r2 + r - 6 = 0
r1 = -3, r2 = 2
an = w1 (-3)n + w2 (2)n
an = - (-3)n + (2)n
37Ch. 4.1, 4.2 & 4.5
☺☺☺☺ Small Exercise ☺☺☺☺
� the recurrence relation: an = - 2an - an-2
� Initial conditions a0 = 5 and a1 = -6
� Characteristic Equation:
� Characteristic Root:
� Therefore,
� Using the initial condition� a0 = 5 = w1
� a1 = -6 = -w1 - w2
� w1 = 5 , w2 = 1
� Therefore,
(r + 1)(r + 1) = 0r2 + 2r + 1 = 0
r1 = -1
an = w1 (-1)n + w2 n (-1)n
an = 5 (-1)n + n (-1)n
38Ch. 4.1, 4.2 & 4.5
Solving k-LiHoReCoCos
� k-LiHoReCoCo:
� Characteristic Equation is:
� TheoremIf there are k distinct roots ri , then the solutions to the recurrence are of the form:
for all n ≥ 0, where the wi are constants
∑=
−=k
i
inin aca1
01
=−∑=
−k
i
ik
i
krcr
∑=
=k
i
n
iin rwa1
an = c1an−1 + c2an−2
r2 − c1r − c2 = 0
an = w1r1n + w2r2
n
2-kiHoReCoCo
39Ch. 4.1, 4.2 & 4.5
Solving k-LiHoReCoCos
Example� Find the solution to the recurrence relation
an = 6an-1 – 11an-2 + 6an-3
with the initial conditions a0 = 2, a1 = 5, and a2 = 15.
� The characteristic equation is:
r3 - 6r2 + 11r – 6 = (r - 1)(r - 2)(r - 3)� The characteristic roots are r = 1, r = 2, and r = 3
� an = w11n + w22
n + w33n
� By using the initial conditions� a0 = 2 = w1 + w2 + w3
� a1 = 5 = w1 + w2 x 2 + w3 x 3
� a2 = 15 = w1 + w2 x 4 + w3 x 9
� Therefore, w1 = 1, w2 = -1 and w3 = 2
� As a result, an = 1 - 2n + 2 x 3n
40Ch. 4.1, 4.2 & 4.5
Solving k-LiHoReCoCoswith same roots
� Let c1, c2, ..., ck be real numbers
� Suppose that the characteristic equation
rk – c1rk-1 – ··· – ck = 0
has t distinct roots r1, r2, ..., rt with multiplicities m1, m2, ..., mt
� i.e. ri appear mi times
� m1 + m2 + ·· · + m t = k
41Ch. 4.1, 4.2 & 4.5
Solving k-LiHoReCoCoswith same roots� A sequence {an} is a solution of the recurrence relation
� If and only if
for n = 0,1,2, ..., where wi,j
are constants for 1 ≤ i ≤ t and 0 ≤ j ≤ m
i-1
nm
mnrnwnwwa1
1
1,11,10,1)...( 1
1
−−+++=
nm
m rnwnww 2
1
1,21,20,2 )...( 2
2
−−++++
n
t
m
mttt rnwnww t
t)...(...
1
1,1,0,
−−+++++
∑ ∑=
−
=
=
t
i
n
i
m
j
j
ji rnwi
1
1
0
,
Special case for k=2, One distinct rootan = w1r0
n + w2nr0n
an = c1an-1 + c2an-2 + ... + ckan-k
No. of distinct roots
Multiplicities for ri
an = (w1 + w2n)r0n
42Ch. 4.1, 4.2 & 4.5
Solving k-LiHoReCoCos with same roots
Example� Find the solution to the recurrence relation
with the initial conditions
� Characteristic equation:
� Roots: −1, −1, −1, 2
� Therefore:
� By initial conditions:
4321253 −−−− +++−=
nnnnnHHHHH
210,1 3210 ==== ,HH,HH
0253234 =−−−+ xxxx
nn
ncncnccH 2)1)((
4
2
321+−++=
0 1 4
1 1 2 3 4
2 1 2 3 4
3 1 2 3 4
1
2 0
2 4 4 1
3 9 8 2
H c c
H c c c c
H c c c c
H c c c c
= + = = − − − + =
= + + + = = − − − + =
9
2,0,
3
1,
9
74321 ==−== cccc
nnn
n nH 29
2)1(
3
1)1(
9
7+−−−=
0)1)(2( 3 =+− xx
nm
mnrnwnwwa1
1
1,11,10,1)...( 1
1
−−+++=
nm
m rnwnww 2
1
1,21,20,2 )...( 2
2
−−++++
n
t
m
mtttrnwnww t
t
)...(...1
1,1,0,
−
−+++++
43Ch. 4.1, 4.2 & 4.5
Solving k-LiHoReCoCos
Summary
� Given : and ai = ci , where i = 1, 2, …, k
1.Characteristic equation:
2. Characteristic Root (r1, r2, …, rk)
1. is the solution of k-LiHoReCoCos
where m i is the multiplicity of ri
2. solve wi by ap = cp
where p = 1, 2, ..., k
01
=−∑=
−k
i
ik
i
krcr
∑=
−=k
i
inin aca1
∑ ∑=
−
=
=
t
i
n
i
m
j
j
jin rnwai
1
1
0
,
∑ ∑=
−
=
=
t
i
p
i
m
j
j
jirpw
i
1
1
0
,
44Ch. 4.1, 4.2 & 4.5
☺☺☺☺ Small Exercise ☺☺☺☺
� What is the solution of the recurrence relation a
n= a
n-1+ a
n-2- a
n-3with initial conditions a
0=
5, a1 = 5 and a2 = -6?
nm
mnrnwnwwa1
1
1,11,10,1)...( 1
1
−−+++=
nm
mrnwnww
2
1
1,21,20,2)...( 2
2
−−++++
n
t
m
mtttrnwnww t
t)...(... 1
1,1,0,
−−+++++
45Ch. 4.1, 4.2 & 4.5
☺☺☺☺ Small Exercise ☺☺☺☺
� the recurrence relation: an = an-1 + an-2 - an-3
� Initial conditions a0 = 5, a1 = 5 and a2 = -6
� Characteristic Equation:
� Characteristic Root:
� Therefore,
� Using the initial condition� a0 = 0 = c1 + c3
� a1 = 8 = c1 + c2 – c3
� a1 = 4 = c1 + 2c2 + c3
� c1 = 3, c2 = -3, c3 = 2
� Therefore,
(r – 1)(r – 1)(r + 1) = 0r3 – r2 – r + 1 = 0
r1 = 1, r2= 1, r3 = -1
an = (c1 + c2 n) (1)n + c3 (-1)n
an = 3 (1 – n) (1)n + 2 (-1)n
46Ch. 4.1, 4.2 & 4.5
Solving LiNoReCoCos
� Linear nonhomogeneous recurrence of degree k with constant coefficients(k-LiNoReCoCos) contain some terms F(n)
that depend only on n but not ai
� General form:
an = c1an−1 + … + cka
n−k + F(n)
Associated Homogeneo us Recurrence Relation
47Ch. 4.1, 4.2 & 4.5
Solving LiNoReCoCos
� If { } is a particular solution of the nonhomogeneous linear recurrence relation with constant coefficients
an = c1an-1 + c2an-2 + ... + ckan-k + F(n)
� Then every solution is of the form { }, where { } is a solution of the associated homogeneous recurrence relation
an = c1an-1 + c2an-2 + ... + ckan-k
)( p
na
)()( h
n
p
n aa +)(h
na
48Ch. 4.1, 4.2 & 4.5
Solving LiNoReCoCos
� Proof
� As { } is a particular solution for LiNoReCoCos
� Suppose that { } is an another solution
� { } is a solution of the associated homogeneous linear recurrence, named { }
� Consequently, for all n.
)(...)()(
22
)(
11
)(nFacacaca
p
knk
p
n
p
n
p
n ++++= −−−
)(...2211 nFbcbcbcb knknnn ++++= −−−
( ) ( ) ( ))()(
222
)(
111
)( ... p
knknk
p
nn
p
nn
p
nnabcabcabcab −−−−−− −++−+−=−
)( p
nn ab −)( h
na)()( h
n
p
nn aab +=
)( p
na
nb
)()(
22
)(
11
)(...
h
knk
h
n
h
n
h
n acacaca −−− +++=
49Ch. 4.1, 4.2 & 4.5
Solving LiNoReCoCos
Example 1
� Find all solutions to an = 3an−1+2n, which solution has a1 = 3?
� Notice this is a 1-LiNoReCoCo.
� Its associated 1-LiHoReCoCo
� an = 3an−1 and root is 3
� Solution is = c3n
� The solutions of LiNoReCoCo are in the form
)(h
na
)()( h
n
p
nn aaa +=
an = 3an−1+2n
a1= 3
Next step
50Ch. 4.1, 4.2 & 4.5
Solving LiNoReCoCos
Example
� If F(n) is a degree-u polynomial in n, a degree-upolynomial should be tried as the particular solution
� Now, F(n) = 2n
� Try = cn + d, c and d are constants
� Solution is: = -n – 3/2
an = 3an-1 + 2n
cn+d = 3(c(n−1)+d) + 2n
(2c+2)n + (3c−2d) = 0
c = −1 and d = −3/2
)( p
na
)( p
na
)( p
na
an = 3an−1+2n
a1= 3
51Ch. 4.1, 4.2 & 4.5
Solving LiNoReCoCos
Example
� Therefore, we have
� By using a1 = 3
� 3 = -1 - 3/2 + 3c
� c = 11/6
� As a result,
)()( h
n
p
nnaaa +=
nh
nca 3
)( =
( ) 3
2
p
na n= − −
33
2
nn c= − − +
6
311
2
3n
nna
⋅++−=
an = 3an−1+2n
a1= 3
52Ch. 4.1, 4.2 & 4.5
� Suppose {an} satisfies the LiNoReCoCo
where ci (i = 1,2,…k) are real numbers and
where b0, b
1, …, b
tand s are real numbers
� When s is not a root of the characteristic equation of the associated li near homogeneous RR, there is a particular solution( ) of the form
� When s is a root of the characteristic equation of the associated linear homogeneo us RR with multiplicity m, there is a particular solution ( ) of the form
)(1
nFacak
i
inin+
= ∑
=−
nt
t
t
t sbnbnbnbnF )...()( 01
1
1 ++++= −−
nt
t
t
tspnpnpnp )...(
01
1
1++++ −
−
nt
t
t
t
m spnpnpnpn )...(01
1
1++++ −
−
Solving LiNoReCoCos
Particular Solution
)( p
na
)( p
na
53Ch. 4.1, 4.2 & 4.5
� F(n) = 3n
� F(n) = n3n
� F(n) = n22n
� F(n) = (n2 + 1)3n
� Consider the associated ho mogeneous RR:
an
= 6an-1
- 9an-2
� Characteristic Equation
� Characteristic Root is 3, of multiplicity m=2
Solving LiNoReCoCos : Particular Solution
Example 1� What form does a particular solution of the linear
nonhomogeneous recurrence relation an = 6an-1 - 9a
n-2+F(n) have when ...
0)3(9622 =−=+− rrr
npn 3)(
0
2
npnpn 3)(
01
2 +
npnpnp 2)(
01
2
2++
npnpnpn 3)(
01
2
2
2 ++
nt
t
t
tspnpnpnp )...(
01
1
1++++ −
−
nt
t
t
t
m spnpnpnpn )...(01
1
1++++ −
−
s is not a root
s is a root
nt
t
t
tsbnbnbnbnF )...()(
01
1
1++++= −
−
54Ch. 4.1, 4.2 & 4.5
Solving LiNoReCoCos : Particular Solution
Example 2� Let an be the sum of the first n positive integers, so that
� an satisfies the linear nonhomogeneous RR
� Associated linear homogeneous RR is an = an-1
� Root is 1. The solution is , c is a constant
� Since F(n) = n = n x (1)n, and s = 1 is a root of degree one of the characteristic equation of the associated linear homogeneous RR
� So the particular solution has the form n(p1n+p0)
nh
n ca )1()( =
∑=
=n
k
n ka1
nt
t
t
t spnpnpnp )...( 01
1
1 ++++ −−
nt
t
t
t
mspnpnpnpn )...( 01
1
1 ++++ −−
s is not a root
s is a root
ntt
tt sbnbnbnbnF )...()( 01
11 ++++= −−
an
= an-1
+ n
55Ch. 4.1, 4.2 & 4.5
Solving LiNoReCoCos : Particular Solution
Example 2
� By solving p1n2 + p0n = p1 (n - 1)2 + p0(n - 1) + n
� We have p0 = p1 =1/2
� Recall,
� By using a1 = 1, so c = 0
� Therefore,
2
)1( +=
nna
n
)()( h
n
p
nnaaa +=
cnnan
++= 2/)1(ca h
n=)(
2/)1()( += nna
p
n
an = an-1 + n( )
1 0( )p
na n p n p= +
56Ch. 4.1, 4.2 & 4.5
Solving Linear Recurrence Relations
Summary� k-LiHoReCoCos with m same roots (without F(x))
� Find the root of characteristic equation
�
� Use initial terms to find alphas
� k-LiNoReCoCos with m same roots (with F(x))
� Find the solution of charac teristic equationof Associated linear homogeneo us RR
� Find the particular solution of LiNoReCoCo using
� Fina lly
� Use initial terms to find alphas
nm
mnrnna1
1
1,11,10,1)...( 1
1
−−+++= ααα
nm
mrnn2
1
1,21,20,2)...( 2
2
−−++++ ααα
n
t
m
mttt rnn t
t)...(...
1
1,1,0,
−−+++++ ααα
nt
t
t
t
mspnpnpnpn )...( 01
1
1 ++++ −−
)()( h
n
p
n aa +
=)( p
na
)(h
na
57Ch. 4.1, 4.2 & 4.5
☺☺☺☺ Small Exercise ☺☺☺☺
� Find all solutions to an = 7an−1+(2n2+2)3n, which solution has a1 = 10?
58Ch. 4.1, 4.2 & 4.5
☺☺☺☺ Small Exercise ☺☺☺☺
� Its associated 1-LiHoReCoCo� an = 7an−1 and root is 7
� Solution is = c7n
� The solutions of 1-LiNoReCoCo are in the form
� Need to do is find one
)(h
na
)()( h
n
p
nn aaa +=
)( p
na
an = 7an−1+(2n2+2)3n
a0 = 10
59Ch. 4.1, 4.2 & 4.5
☺☺☺☺ Small Exercise ☺☺☺☺
� Now, F(n) = (2n2+2)3n
� = (an2 + bn + c)3n
an = 7an-1 + (2n2+2)3n
)( p
na
(an2+bn+c)3n = 7(a(n-1)2+b(n-1)+c)3n-1 + (2n2+2)3n
3an2+3bn+3c = 7an2 - 14an + 7a + 7bn – 7b + 7c + 6n2 + 6
0 = n2(4a+6)+n(4b-14a) +(4c +7a-7b+6 )
4c +7a-7b+6=04b-14a=04a+6=0
a=-3/2 b=-21/4 c=-129/16
np
n nna 3)6/1294/212/3(2)( −−−=
an = 7an−1+(2n2+2)3n
a0 = 10
0 = 4an2 - 14an + 7a + 4bn – 7b + 4c + 6n2 + 6
60Ch. 4.1, 4.2 & 4.5
☺☺☺☺ Small Exercise ☺☺☺☺
� Therefore, we have
� By using a0 = 10
� a0 = 10 = -129/6 + c
� c = 189/6
� As a result,
)()( h
n
p
nn aaa +=nh
nca 7
)( =nn cnn 73)6/1294/212/3( 2 +−−−=
6/71893)6/1294/212/3(2 nn
nnna ⋅+−−−=
np
nnna 3)6/1294/212/3(
2)( −−−=
an = 7an−1+(2n2+2)3n
a0 = 10
61Ch. 4.1, 4.2 & 4.5
Generating Functions
� Generating functions ( G(x) ) are used to represent sequences efficiently by coding the terms of a sequence as coefficients of powers of a variable x in a formal power series
� Generating function for the sequence
a0, a1, a2, …, ak, … of real numbers is the infinite series
0 1
0
0 1( )k
k k
k kG x x x xa a a xa∞
=
= + + + + =∑⋯ ⋯
62Ch. 4.1, 4.2 & 4.5
Example
� What is the generating function for the
following sequence?
� {0, 2, …, 2k,…}
� {1, 1, 1, 1, 1}
∑∞
=0
2k
kkx
∑=
4
0k
kx1 + x + x2 + x3 + x4 =
0 + 2x + … + 2k · xk + … =
63Ch. 4.1, 4.2 & 4.5
Useful Facts About Power Series
�
is generating functionof the sequence 1,1,1,1, ...
�
is generating functionof the sequence 1, a, a2, a3, ...
� Recall, in Chapter 2.4,
( )x
xf−
=1
1 ( )ax
xf−
=1
1
∑∞
=0k
kx ∑
∞
=0k
kkxa
x−=
1
1
ax−=
1
1
64Ch. 4.1, 4.2 & 4.5
Useful Facts About Power Series
� Given: ( ) ∑∞
=
=0k
k
k xaxf ( ) ∑∞
=
=0k
k
k xbxg
( ) ( )=+ xgxf
( ) ( )=xgxf
( )∑∞
=
+0k
k
kk xba
( )∑∑∞
= =−=
0 0k
k
j
k
jkj xba
∑∑∞
=
∞
= 00 k
k
k
k
k
k xbxa
...)...)((1
1
0
0
1
1
0
0 ++++= xbxbxaxa
+++ )()( 0110
1
00
0babaxbax=
...)( 021120
2 +++ bababax00ba 0110 baba +
021120 bababa ++
65Ch. 4.1, 4.2 & 4.5
Useful Facts About Power Series
Example
� Let ,
� Find the coefficients a0, a1, a2, ... in the expansion
∑∞
==
0)(
k
k
kxaxh
2)1(
1)(
xxh
−=
( ) ∑∞
=
=−
=01
1
k
kx
xxf
( ) ∑∞
=
=0k
k
kxaxf ( ) ∑
∞
=
=0k
k
kxbxg
( ) ( ) ( )∑∞
=
+=+0k
k
kkxbaxgxf
( ) ( ) ( )∑∑∞
= =−=
0 0k
k
j
k
jkjxbaxgxf
)(xh)1(
1
)1(
1
xx −−=
2)1(
1
x−=
0 0
kk
k j
x∞
= =
=∑∑
∑∞
=
+=0
)1(k
kxkak=k+1
0 0
k k
k k
x x∞ ∞
= =
= ∑ ∑
66Ch. 4.1, 4.2 & 4.5
Extended Binomial Coefficient
� Recall, the binomial coefficient,
where n is a nonnegative integer and r is an integer with 0 ≤ r ≤ n
� The extended binomial coefficients is defined by replacing nonngeative integer n by real number u
!
)1)...(1(
r
rnnn +−−=
)!(!
!
rnr
n
−=
r
n
67Ch. 4.1, 4.2 & 4.5
Extended Binomial Coefficient
Example
� Find the values of the extended binomial coefficients
−
3
2
3
2/1
!
)1)...(1(
r
rnnn
r
n +−−=
3
5 ( ) ( )123
25155
⋅⋅
−⋅−⋅= 10=
( ) ( )123
22122
⋅⋅
−−⋅−−⋅−= 4−=
( ) ( ) ( )123
22/112/12/1
⋅⋅
−−⋅−−⋅−=
16
1=
(Standard binomial coefficients)
68Ch. 4.1, 4.2 & 4.5
Extended Binomial Coefficient
� When the top parameter is a negative integer, the extended binomial coefficient can be expressed in terms of an ordinary binomial coefficient
−
r
n ( )( ) ( )!
11
r
rnnn +−−⋅⋅⋅−−−=
( ) ( )( ) ( )!
111
r
rnnnr −+⋅⋅⋅+−
=
( ) ( )( ) ( )!
211
r
nrnrnr ⋅⋅⋅−+−+−
=
( )
−+−=
r
rnr 11
69Ch. 4.1, 4.2 & 4.5
Extended Binomial Theorem
� Recall, the ordinary binomial theorem
� Extended binomial theorem
where x is a real number with |x|<1, u is a real number
k
k
u xk
ux ∑
∞
=
=+
0
)1(
( )nyx+ jjn
n
j
yxj
n −
=∑
=
0
70Ch. 4.1, 4.2 & 4.5
Extended Binomial Theorem
Example
� Find the generating functions for (1+x)-n and (1-x)-n, where n is a positive integer, using the extended binomial theorem.
nx
−− )1(
nx
−+ )1(
( )kk
xk
n−
−=∑
∞
=0
k
k
xk
n∑∞
=
−=
0
∑∞
=
−+=0
),1(k
kxkknC
∑∞
=
−+−=0
),1()1(k
kkxkknC
−
r
n ( )
−+−=
r
rnr 11∑
∞
=
=0
)(k
k
k xaxG
ak
k
k
u xk
ux ∑
∞
=
=+
0
)1(
71Ch. 4.1, 4.2 & 4.5
Counting Problems and Generating Functions� How can we solve the counting problems,
including the recurrence relation, by using the Generating Functions?
0 1
0
0 1( )k
k k
k kG x x x xa a a xa∞
=
= + + + + =∑⋯ ⋯
72Ch. 4.1, 4.2 & 4.5
Counting Problems and Generating Functions
Example 1� Find the number of solutions of e1 + e2 + e3 = n when n = 17,
where e1 ,e2, e3 are nonnegative integers with 2 ≤≤≤≤ e1 ≤≤≤≤ 5, 3 ≤≤≤≤ e2 ≤≤≤≤ 6, 4 ≤≤≤≤ e3 ≤≤≤≤ 7
� By considering this generating function for the sequence {an}, where an is the number of solution for n
� As n = 17, a17, which is the coefficient of x17, is the solution
� Answer is 3
=∑∞
=0k
k
kxa
⋅+++ )( 5432 xxxx
⋅+++ )(6543
xxxx
)(7654
xxxx +++
73Ch. 4.1, 4.2 & 4.5
Counting Problems and Generating Functions
Example 2
� In how many different ways can eight identical cookies be distributed among three distinct childrenif each child receives at least two cookies and no more than four cookies?
� By considering this generating function for the sequence {an}, where an is the number of solution for n
� The coefficient of x8 is 6
))()((432432432
xxxxxxxxx ++++++=∑∞
=0k
k
kxa
74Ch. 4.1, 4.2 & 4.5
Counting Problems and Generating Functions
Example 3
� Use generating functions to determine the number of ways to insert tokens worth $1, $2, and $5 into a vending machine to pay for an item that costs r dollars in both the cases when the order in which
� the order of inserting tokens does not matter
� the order does matter
� For example,
� Item that costs $3
� Order does not matter (2)
� Three $1 tokens
� $1 token and $2 token
� Order does matter (3)
� Three $1 tokens
� $1 token and a $2 token
� $2 token and a $1 token
75Ch. 4.1, 4.2 & 4.5
Counting Problems and Generating Functions
Example 3
� Order does not matter
� By considering:
� $1:
� $2:
� $5:
� The answer is the coefficient of xr in
...)(321 +++ xxx
...)(642 +++ xxx
...)( 15105 +++ xxx
1 2 3
2 4 6
5 10 15
( . ..)( . ..)( ...)
x x xx x xx x x
+ + + ⋅+ + + ⋅+ + +
=∑∞
=0k
k
kxa
76Ch. 4.1, 4.2 & 4.5
Counting Problems and Generating Functions
Example 3
� Order does matter
� The number of ways to insert n tokens to produce a total of r dollars is the coefficient of xr in
� Answer is the coefficient of xr in
nxxx )( 521 ++
xx
k
k
−=∑
∞
= 1
1
0
1 2 5
0
( )n
n
x x x∞
=
= + +∑
)(1
152
xxx ++−=
521
1
xxx −−−=
1 2 5 1 2 5 21 ( ) ( ) ...x x x x x x+ + + + + + +
E.g. r = 5
1 2 5 2( )x x x+ +
1 2 5 3( )x x x+ +1 2 5 4
( )x x x+ +1 2 5 5
( )x x x+ +
11 2 5
( )x x x+ +
0
3
4
1
Coef. of x5
=∑∞
=0k
k
kxa
77Ch. 4.1, 4.2 & 4.5
Counting Problems and Generating Functions
Example 4� Use generating functions to find the number of ways
to select r objects of n different kinds if we must select at least one object of each kind.
� Each kind of objects contributes
to generating function G(x) for the sequence {xr}, where xr is the number of ways to select r objects of n different kinds
� Hence, there are n kinds:
At least one object of each kind
nxxx ...)( 321 +++=
...)(321 +++ xxx
)(xGnn xxx ...)1( 21 +++=
n
n
x
x
)1( −=
78Ch. 4.1, 4.2 & 4.5
Counting Problems and Generating Functions
Example 4
)(xGnn xx )1/( −=
(1 )n nx x −= −
r
r
nx
r
nx )(
0
−
−= ∑
∞
=
rr
r
rnxrrnCx )()1(),1()1(
0
−−+−= ∑∞
=
rn
r
xrrnC+
∞
=∑ −+= )(),1(
0
t
nt
xnttC )(),1(∑∞
=
−−=
r
nr
xnrrC )(),1(∑∞
=
−−=
There are C(r - 1, r - n) ways
to select r objects of n different kinds if we must
select at least one object of
each kind.
Extended binomial theore m
k
k
ux
k
ux ∑
∞
=
=+
0
)1(
−
r
n( )
−+−=
r
rnr 11
79Ch. 4.1, 4.2 & 4.5
Counting Problems and Generating Functions
Example 5
� Solve the recurrence relation ak = 3ak-1 for k=1, 2, 3,
… and initial condition a0=2
� Let G(x) be the generating function for the sequence {ak}, that is
∑∞
=
=0
)(k
k
kxaxG
0
1
( ) k
k
k
G x a a x∞
=
= +∑
0 1
1
( ) 3 k
k
k
G x a a x∞
−=
= +∑1
0 1
1
( ) 3 k
k
k
G x a x a x∞
−−
=
= + ∑
0( ) 3 ( )G x a xG x= +
2( )
1 3G x
x=
−
0
( ) 2 3k k
k
G x x∞
=
= ⋅∑
∑∞
=0k
kk xaax−
=1
1
2 3k
ka = ⋅
80Ch. 4.1, 4.2 & 4.5
Counting Problems and Generating Functions
Example 6
� The sequence {an} satisfies the recurrence relation
an
= 8an-1 + 10n -1
and the initial condition a1 = 9
� Use generating functions to find an explicit formula for an
81Ch. 4.1, 4.2 & 4.5
Counting Problems and Generating Functions
Example 6
a0 = 1
a1
= 8a0
+ 101-1 = 90
( )n
n
n
G x a x∞
=
=∑
0
1
( ) n
n
n
G x a a x∞
=
= +∑1
0 1
1
( ) (8 10 )n n
n
n
G x a a x∞
−−
=
= + +∑1
0 1
1 1
( ) 8 10n n n
n
n n
G x a a x x∞ ∞
−−
= =
= + +∑ ∑ 0
1
1
k k
k
a xax
∞
=
=−
∑
1 1 1
0 1
1 0
( ) 8 10n n n
n
n n
G x a x a x x x∞ ∞
− − −−
= =
= + +∑ ∑
0( ) 8 ( )
1 10
xG x a xG x
x= + +
−
an = 8an-1 + 10n-1 a1 = 9
82Ch. 4.1, 4.2 & 4.5
Counting Problems and Generating Functions
Example 6
1 9( )
(1 8 )(1 10 )
xG x
x x
−=
− −
1 1 1
2 1 8 1 10x x
= + − −
0 0
18 10
2
n n n n
n n
x x∞ ∞
= =
= +
∑ ∑
( )0
18 10
2
n n n
n
x∞
=
= +∑ ( )18 10
2
n n
na = +
0( ) 8 ( )1 10
xG x a xG x
x= + +
−a0 = 1
0
1
1
k k
k
a xax
∞
=
=−
∑