Introductory Combinatorics

Richard A. Brualdi

Chapter 7

Recurrence Relations and Generating Functions - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $f_{0}, f_{1}, f_{2}, \ldots, f_{n}, \ldots$ denote the Fibonacci sequence. By evaluating each of the following expressions for small values of $n$, conjecture a general formula and then prove it, using mathematical induction and the Fibonacci recurrence:
(a) $f_{1}+f_{3}+\cdots+f_{2 n-1}$
(b) $f_{0}+f_{2}+\cdots+f_{2 n}$
(c) $f_{0}-f_{1}+f_{2}-\cdots+(-1)^{n} f_{n}$
(d) $f_{0}^{2}+f_{1}^{2}+\cdots+f_{n}^{2}$

Manisha Sarker

Manisha Sarker

Numerade Educator

Problem 2

Prove that the $n$ th Fibonacci number $f_{n}$ is the integer that is closest to the number
$$\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^{n}$$

Chris Trentman

Numerade Educator

Problem 3

Prove the following about the Fibonacci numbers:
(a) $f_{n}$ is even if and only if $n$ is divisible by 3 .
(b) $f_{n}$ is divisible by 3 if and only if $n$ is divisible by 4 .
(c) $f_{n}$ is divisible by 4 if and only if $n$ is divisible by 6 .

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 4

Prove that the Fibonacci sequence is the solution of the recurrence relation
$$a_{n}=5 a_{n-4}+3 a_{n-5}, \quad(n \geq 5)$$
where $a_{0}=0, a_{1}=1, a_{2}=1, a_{3}=2$, and $a_{4}=3 .$ Then use this formula to show that the Fibonacci numbers satisfy the condition that $f_{n}$ is divisible by 5 if and only if $n$ is divisible by 5 .

Bryan Lynn

Numerade Educator

Problem 5

By examining the Fibonacci sequence, make a conjecture about when $f_{n}$ is divisible by 7 and then prove your conjecture.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 6

$*$ Let $m$ and $n$ be positive integers. Prove that if $m$ is divisible by $n$, then $f_{m}$ is divisible by $f_{n}$

Manisha Sarker

Numerade Educator

Problem 7

Let $m$ and $n$ be positive integers whose greatest common divisor is $d$. Prove that the greatest common divisor of the Fibonacci numbers $f_{m}$ and $f_{n}$ is the Fibonacci number $f_{d}$.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 8

Consider a 1-by-n chessboard. Suppose we color each square of the chessboard with one of the two colors red and blue. Let $h_{n}$ be the number of colorings in which no two squares that are colored red are adjacent. Find and verify a recurrence relation that $h_{n}$ satisfies. Then derive a formula for $h_{n}$.

Chris Trentman

Numerade Educator

Problem 9

Let $h_{n}$ equal the number of different ways in which the squares of a 1 -by-n chessboard can be colored, using the colors red, white, and blue so that no two squares that are colored red are adjacent. Find and verify a recurrence relation that $h_{n}$ satisfies. Then find a formula for $h_{n}$.

Chris Trentman

Numerade Educator

Problem 10

Suppose that, in his problem, Fibonacci had placed two pairs of rabbits in the enclosure at the beginning of a year. Find the number of pairs of rabbits in the enclosure after one year. More generally, find the number of pairs of rabbits in the enclosure after $n$ months.

AG

Ankit Gupta

Numerade Educator

Problem 11

The Lucas numbers $l_{0}, l_{1}, l_{2}, \ldots, l_{n} \ldots$ are defined using the same recurrence relation defining the Fibonacci numbers, but with different initial conditions:
$$L_{n}=l_{n-1}+l_{n-2},(n \geq 2), l_{0}=2, l_{1}=1$$
Prove that
(a) $l_{n}=f_{n-1}+f_{n+1}$ for $n \geq 1$
(b) $l_{0}^{2}+l_{1}^{2}+\cdots+l_{n}^{2}=l_{n} l_{n+1}+2$ for $n \geq 0$

Chris Trentman

Numerade Educator

Problem 12

Let $h_{0}, h_{1}, h_{2}, \ldots, h_{n}, \ldots$ be the sequence defined by
$$h_{n}=n^{3},(n \geq 0)$$
Show that $h_{n}=h_{n-1}+3 n^{2}-3 n+1$ is the recurrence relation for the sequence.

Farnood Ensan

Numerade Educator

Problem 13

Determine the generating function for each of the following sequences:
(a) $c^{0}=1, c, c^{2}, \ldots, c^{n}, \ldots$
(b) $1,-1,1,-1, \ldots,(-1)^{n}, \ldots$
(c) $\left(\begin{array}{l}\alpha \\ 0\end{array}\right),-\left(\begin{array}{c}\alpha \\ 1\end{array}\right),\left(\begin{array}{c}\alpha \\ 2\end{array}\right), \ldots,(-1)^{n}\left(\begin{array}{c}\alpha \\ n\end{array}\right), \ldots,(\alpha$ is a real number $)$
(d) $1, \frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{n}, \ldots$
(e) $1,-\frac{1}{11}, \frac{1}{21}, \ldots,(-1)^{n} \frac{1}{n}, \ldots$

Chris Trentman

Numerade Educator

Problem 14

Let $S$ be the multiset $\left\{\infty \cdot e_{1}, \infty \cdot e_{2}, \infty \cdot e_{3}, \infty \cdot e_{4}\right\}$. Determine the generating function for the sequence $h_{0}, h_{1}, h_{2}, \ldots, h_{n}, \ldots$, where $h_{n}$ is the number of $n$ combinations of $S$ with the following added restrictions:
(a) Each $e_{i}$ occurs an odd number of times.
(b) Each $e_{i}$ occurs a multiple-of-3 number of times.
(c) The element $e_{1}$ does not occur, and $e_{2}$ occurs at most once.
(d) The element $e_{1}$ occurs 1,3, or 11 times, and the element $e_{2}$ occurs 2,4, or 5 times.
(e) Each $e_{i}$ occurs at least 10 times.

WZ

Wen Zheng

Numerade Educator

Problem 15

Determine the generating function for the sequence of cubes
$$0,1,8, \ldots, n^{3}, \ldots$$

Chris Trentman

Numerade Educator

Problem 16

Formulate a combinatorial problem for which the generating function is
$$\left(1+x+x^{2}\right)\left(1+x^{2}+x^{4}+x^{6}\right)\left(1+x^{2}+x^{4}+\cdots\right)\left(x+x^{2}+x^{3}+\cdots\right)$$

Chris Trentman

Numerade Educator

Problem 17

Determine the generating function for the number $h_{n}$ of bags of fruit of apples, oranges, bananas, and pears in which there are an even number of apples, at most two oranges, a multiple of three number of bananas, and at most one pear. Then find a formula for $h_{n}$ from the generating function.

Chris Trentman

Numerade Educator

Problem 18

Determine the generating function for the number $h_{n}$ of nonnegative integral solutions of
$$2 e_{1}+5 c_{2}+e_{3}+7 e_{4}=n$$

Chris Trentman

Numerade Educator

Problem 19

Let $h_{0}, h_{1}, h_{2}, \ldots, h_{n}, \ldots$ be the sequence defined by $h_{n}=\left(\begin{array}{c}n \\ 2\end{array}\right),(n \geq 0)$. Deter-
mine the generating function for the sequence.

Aman Gupta

Numerade Educator

Problem 20

Let $h_{0}, h_{1}, h_{2}, \ldots, h_{n}, \ldots$ be the sequence defined by $h_{n}=\left(\begin{array}{c}n \\ 3\end{array}\right),(n \geq 0)$. Deter-
mine the generating function for the sequence.

WZ

Wen Zheng

Numerade Educator

Problem 21

Let $h_{n}$ denote the number of regions into which a convex polygonal region with $n+2$ sides is divided by its diagonals, assuming no three diagonals have a common point. Define $h_{0}=0 .$ Show that
$$h_{n}=h_{n-1}+\left(\begin{array}{c}n+1 \\
3\end{array}\right)+n, \quad(n \geq 1)$$
Then determine the generating function and obtain a formula for $h_{n}$.

JW

Julian Wong

Numerade Educator

Problem 22

Determine the exponential generating function for the sequence of factorials:
$0 !, 11,2 !, 3 !, \ldots, n !, \ldots$

Chris Trentman

Numerade Educator

Problem 23

Let $\alpha$ be a real number. Let the sequence $h_{0}, h_{1}, h_{2}, \ldots, h_{n}, \ldots$ be defined by $h_{0}=1$, and $h_{n}=\alpha(\alpha-1) \cdots(\alpha-n+1),(n \geq 1)$. Determine the exponential generating function for the sequence.

WZ

Wen Zheng

Numerade Educator

Problem 24

Let $S$ denote the multiset $\left\{\infty \cdot e_{1}, \infty \cdot e_{2}, \ldots, \infty \cdot e_{k}\right\} .$ Determine the exponential generating function for the sequence $h_{0}, h_{1}, h_{2}, \ldots, h_{n}, \ldots$, where $h_{0}=1$ and, for $n \geq 1$,
(a) $h_{n}$ equals the number of $n$ -permutations of $S$ in which each object occurs an odd number of times.
(b) $h_{n}$ equals the number of $n$ -permutations of $S$ in which each object occurs at least four times.
(c) $h_{n}$ equals the number of $n$ -permutations of $S$ in which $e_{1}$ occurs at least once, $e_{2}$ occurs at least twice, $\ldots, e_{k}$ occurs at least $k$ times.
(d) $h_{n}$ equals the number of $n$ -permutations of $S$ in which $e_{1}$ occurs at most once, $e_{2}$ occurs at most twice, $\ldots, e_{k}$ occurs at most $k$ times.

WZ

Wen Zheng

Numerade Educator

Problem 25

Let $h_{n}$ denote the number of ways to color the squares of a 1 -by- $n$ board with the colors red, white, blue, and green in such a way that the number of squares colored red is even and the number of squares colored white is odd. Determine the exponential generating function for the sequence $h_{0}, h_{1}, \ldots, h_{n}, \ldots$, and then find a simple formula for $h_{n}$.

Aman Gupta

Numerade Educator

Problem 26

Determine the number of ways to color the squares of a 1-by-n chessboard, using the colors red, blue, green, and orange if an even number of squares is to be colored red and an even number is to be colored green.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 27

Determine the number of $n$ -digit numbers with all digits odd, such that 1 and 3 each occur a nonzero, even number of times.

Hunza Gilgit

Numerade Educator

Problem 28

Determine the number of $n$ -digit numbers with all digits at least 4 , such that 4 and 6 each occur an even number of times, and 5 and 7 each occur at least once, there being no restriction on the digits 8 and $9 .$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 29

We have used exponential generating functions to show that the number $h_{n}$ of $n$ -digit numbers with each digit odd, where the digits 1 and 3 occur an even number of times, satisfies the formula
$$h_{n}=\frac{5^{n}+2 \times 3^{n}+1}{4}, \quad(n \geq 0)$$
Obtain an alternative derivation of this formula.

Chris Trentman

Numerade Educator

Problem 30

We have used exponential generating functions to show that the number $h_{n}$ of ways to color the squares of a 1-by-n board with the colors red, white, and blue, where the number of red squares is even and there is at least one blue square, satisfies the formula
$$h_{n}=\frac{3^{n}-2^{n}+1}{2}, \quad(n \geq 1)$$
with $h_{0}=0 .$ Obtain an alternative derivation of this formula by finding a recurrence relation satisfied by $h_{n}$ and then solving the recurrence relation.

Chris Trentman

Numerade Educator

Problem 31

Solve the recurrence relation $h_{n}=4 h_{n-2},(n \geq 2)$ with initial values $h_{0}=0$ and $h_{1}=1$.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 32

Solve the recurrence relation $h_{n}=(n+2) h_{n-1},(n \geq 1)$ with initial value $h_{0}=2$.

Chris Trentman

Numerade Educator

Problem 33

Solve the recurrence relation $h_{n}=h_{n-1}+9 h_{n-2}-9 h_{n-3},(n \geq 3)$ with initial values $h_{0}=0, h_{1}=1$, and $h_{2}=2$.

Chris Trentman

Numerade Educator

Problem 34

Solve the recurrence relation $h_{n}=8 h_{n-1}-16 h_{n-2},(n \geq 2)$ with initial values $h_{0}=-1$ and $h_{1}=0 .$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 35

Solve the recurrence relation $h_{n}=3 h_{n-2}-2 h_{n-3}, \quad(n \geq 3)$ with initial values $h_{0}=1, h_{1}=0$, and $h_{2}=0$

Chris Trentman

Numerade Educator

Problem 36

Solve the recurrence relation $h_{n}=5 h_{n-1}-6 h_{n-2}-4 h_{n-3}+8 h_{n-4},(n \geq 4)$ with initial values $h_{0}=0, h_{1}=1, h_{2}=1$, and $h_{3}=2$.

Chris Trentman

Numerade Educator

Problem 37

Determine a recurrence relation for the number $a_{n}$ of ternary strings (made up of $0 \mathrm{~s}, 1 \mathrm{~s}$, and $2 \mathrm{~s}$ ) of length $n$ that do not contain two consecutive 0's or two consecutive 1s. Then find a formula for $a_{n}$.

Bryan Lynn

Numerade Educator

Problem 38

Solve the following recurrence relations by examining the first few values for a formula and then proving your conjectured formula by induction.
(a) $h_{n}=3 h_{n-1}, \quad(n \geq 1) ; h_{0}=1$
(b) $h_{n}=h_{n-1}-n+3,(n \geq 1) ; h_{0}=2$
(c) $h_{n}=-h_{n-1}+1,(n \geq 1) ; h_{0}=0$
(d) $h_{n}=-h_{n-1}+2,(n \geq 1) ; h_{0}=1$
(e) $h_{n}=2 h_{n-1}+1, \quad(n \geq 1) ; h_{0}=1$

Farnood Ensan

Numerade Educator

Problem 39

Let $h_{n}$ denote the number of ways to perfectly cover a 1 -by- $n$ board with monominoes and dominoes in such a way that no two dominoes are consecutive. Find, but do not solve, a recurrence relation and initial conditions satisfied by $h_{n}$.

Bryan Lynn

Numerade Educator

Problem 40

Let $a_{n}$ equal the number of ternary strings of length $n$ made up of $0 \mathrm{~s}, .1 \mathrm{~s}$, and 2s, such that the substrings $00,01,10$, and 11 never occur. Prove that
$$a_{n}=a_{n-1}+2 a_{n-2}, \quad(n \geq 2)$$
with $a_{0}=1$ and $a_{1}=3 .$ Then find a formula for $a_{n}$.

Bryan Lynn

Numerade Educator

Problem 41

$^{*}$ Let $2 n$ equally spaced points be chosen on a circle. Let $h_{n}$ denote the number of ways to join these points in pairs so that the resulting line segments do not int

Ashley High

Numerade Educator

Problem 42

Solve the nonhomogeneous recurrence relation
$$\begin{array}{l}h_{n}=4 h_{n-1}+4^{n}, \quad(n \geq 1) \\
h_{0}=3 .\end{array}$$

Bryan Lynn

Numerade Educator

Problem 43

Solve the nonhomogeneous recurrence relation
$$h_{n}=4 h_{n-1}+3 \times 2^{n}, \quad(n \geq 1)$$
$$h_{0}=1$$

Bryan Lynn

Numerade Educator

Problem 44

Solve the nonhomogeneous recurrence relation
$$\begin{array}{l}h_{n}=3 h_{n-1}-2, \quad(n \geq 1) \\h_{0}=1 .\end{array}$$

Chris Trentman

Numerade Educator

Problem 45

Solve the nonhomogeneous recurrence relation
$$\begin{array}{l}h_{n}=2 h_{n-1}+n, \quad(n \geq 1) \\h_{0}=1
\end{array}$$

Chris Trentman

Numerade Educator

Problem 46

Solve the nonhomogeneous recurrence relation
$$\begin{array}{l}
h_{n}=6 h_{n-1}-9 h_{n-2}+2 n, \quad(n \geq 2) \\h_{0}=1 \\h_{1}=0
\end{array}$$

Bryan Lynn

Numerade Educator

Problem 47

Solve the nonhomogeneous recurrence relation
$$\begin{array}{l}h_{n}=4 h_{n-1}-4 h_{n-2}+3 n+1, \quad(n \geq 2) \\h_{0}=1 \\
h_{1}=2\end{array}$$

Bryan Lynn

Numerade Educator

Problem 48

Solve the following recurrence relations by using the method of generating functions as described in Section $7.4$ :
(a) $h_{n}=4 h_{n-2},(n \geq 2) ; h_{0}=0, h_{1}=1$
(b) $h_{n}=h_{n-1}+h_{n-2},(n \geq 2) ; h_{0}=1, h_{1}=3$
(c) $h_{n}=h_{n-1}+9 h_{n-2}-9 h_{n-3},(n \geq 3) ; h_{0}=0, h_{1}=1, h_{2}=2$
(d) $h_{n}=8 h_{n-1}-16 h_{n-2},(n \geq 2) ; h_{0}=-1, h_{1}=0$
(e) $h_{n}=3 h_{n-2}-2 h_{n-3},(n \geq 3) ; h_{0}=1, h_{1}=0, h_{2}=0$
(f) $h_{n}=5 h_{n-1}-6 h_{n-2}-4 h_{n-3}+8 h_{n-4},(n \geq 4) ; h_{0}=0, h_{1}=1, h_{2}=1, h_{3}=2$

Chris Trentman

Numerade Educator

Problem 49

$(q$ -binomial theorem) Prove that
$$(x+y)(x+q y)\left(x+q^{2} y\right) \cdots\left(x+q^{n-1} y\right)=\sum_{k=0}^{n}\left(\begin{array}{l}n \\k\end{array}\right)_{q} x^{n-k} y^{k},$$
where
$$n !_{q}=\frac{\prod_{j=1}^{n}\left(1-q^{j}\right)}{(1-q)^{n}}$$
is the $q$ -factorial (cf. Theorem $7.2 .1$ replacing $q$ in $(7.14)$ with $x$ ) and$$\left(\begin{array}{l}n \\k\end{array}\right)_{q}=\frac{n !_{q}}{k !_{q}(n-k) !_{q}}$$
is the $q$ -binomial coefficient.

Nick Johnson

Numerade Educator

Problem 50

Call a subset $S$ of the integers $\{1,2, \ldots, n\}$ extraordinary provided its smallest integer equals its size:
$$\min \{x: x \in S\}=|S|$$
For example, $S=\{3,7,8\}$ is extraordinary. Let $g_{n}$ be the number of extraordinary subsets of $\{1,2, \ldots, n\}$. Prove that
$$g_{n}=g_{n-1}+g_{n-2}, \quad(n \geq 3)$$
with $g_{1}=1$ and $g_{2}=1$.

Carson Merrill

Numerade Educator

Problem 51

Solve the recurrence relation
$$\begin{array}{l}h_{n}=3 h_{n-1}-4 n, \quad(n \geq 1) \\h_{0}=2\end{array}$$
from Section $7.6$ using generating functions.

Chris Trentman

Numerade Educator

Problem 52

Solve the following two recurrence relations:
(a) $h_{n}=2 h_{n-1}+5^{n},(n \geq 1)$ with $h_{0}=3$
(b) $h_{n}=5 h_{n-1}+5^{n},(n \geq 1)$ with $h_{0}=3$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 53

Suppose you deposit $$\$ 500$$ in a bank account that pays $6 \%$ interest at the end of each year (compounded annually). Thereafter, at the beginning of each year you deposit $$\$ 100 .$$ Let $h_{n}$ be the amount in your account after n years $\left(\right.$ so $\left.h_{0}=\$ 500\right)$. Determine the generating function $$g(x)=h_{0}+h_{1} x+\cdots+h_{n} x^{n}+\cdots$$ and then a formula for $h_{n}$.

Laura Skalaski

Numerade Educator