1. 15-251
Great Theoretical Ideas in
Computer Science

2. Recurrences, Fibonacci Numbers
and Continued Fractions
Lecture 9, September 23, 2008

3. Happy Autumnal Equinox
http://apod.nasa.gov/apod/ap080922.html

4. Leonardo Fibonacci

5. Leonardo Fibonacci
In 1202, Fibonacci proposed a problem
about the growth of rabbit populations

6. Rabbit Reproduction

7. Rabbit Reproduction
A rabbit lives forever

8. Rabbit Reproduction
A rabbit lives forever
The population starts as single newborn pair

9. Rabbit Reproduction
A rabbit lives forever
The population starts as single newborn pair
Every month, each productive pair begets
a new pair which will become productive
after 2 months old

10. Rabbit Reproduction
A rabbit lives forever
The population starts as single newborn pair
Every month, each productive pair begets
a new pair which will become productive
after 2 months old
Fn= # of rabbit pairs at the beginning of
the nth month

11. month 1 2 3 4 5 6 7
rabbits
Rabbit Reproduction
A rabbit lives forever
The population starts as single newborn pair
Every month, each productive pair begets
a new pair which will become productive
after 2 months old
Fn= # of rabbit pairs at the beginning of
the nth month

12. month 1 2 3 4 5 6 7
rabbits
Rabbit Reproduction
A rabbit lives forever
The population starts as single newborn pair
Every month, each productive pair begets
a new pair which will become productive
after 2 months old
Fn= # of rabbit pairs at the beginning of
the nth month
1

13. month 1 2 3 4 5 6 7
rabbits
Rabbit Reproduction
A rabbit lives forever
The population starts as single newborn pair
Every month, each productive pair begets
a new pair which will become productive
after 2 months old
Fn= # of rabbit pairs at the beginning of
the nth month
1 1

14. month 1 2 3 4 5 6 7
rabbits
Rabbit Reproduction
A rabbit lives forever
The population starts as single newborn pair
Every month, each productive pair begets
a new pair which will become productive
after 2 months old
Fn= # of rabbit pairs at the beginning of
the nth month
1 1 2

15. month 1 2 3 4 5 6 7
rabbits
Rabbit Reproduction
A rabbit lives forever
The population starts as single newborn pair
Every month, each productive pair begets
a new pair which will become productive
after 2 months old
Fn= # of rabbit pairs at the beginning of
the nth month
1 1 3
2

16. month 1 2 3 4 5 6 7
rabbits
Rabbit Reproduction
A rabbit lives forever
The population starts as single newborn pair
Every month, each productive pair begets
a new pair which will become productive
after 2 months old
Fn= # of rabbit pairs at the beginning of
the nth month
1 1 3 5
2

17. month 1 2 3 4 5 6 7
rabbits
Rabbit Reproduction
A rabbit lives forever
The population starts as single newborn pair
Every month, each productive pair begets
a new pair which will become productive
after 2 months old
Fn= # of rabbit pairs at the beginning of
the nth month
1 1 3 5
2 8

18. month 1 2 3 4 5 6 7
rabbits
Rabbit Reproduction
A rabbit lives forever
The population starts as single newborn pair
Every month, each productive pair begets
a new pair which will become productive
after 2 months old
Fn= # of rabbit pairs at the beginning of
the nth month
1 1 3 5
2 8 13

19. Fibonacci Numbers
month 1 2 3 4 5 6 7
rabbits 1 1 3 5
2 8 13

20. Fibonacci Numbers
Stage 0, Initial Condition, or Base Case:
Fib(1) = 1; Fib (2) = 1
month 1 2 3 4 5 6 7
rabbits 1 1 3 5
2 8 13

21. Fibonacci Numbers
Stage 0, Initial Condition, or Base Case:
Fib(1) = 1; Fib (2) = 1
Inductive Rule:
For n>3, Fib(n) =
month 1 2 3 4 5 6 7
rabbits 1 1 3 5
2 8 13

22. Fibonacci Numbers
Stage 0, Initial Condition, or Base Case:
Fib(1) = 1; Fib (2) = 1
Inductive Rule:
For n>3, Fib(n) = Fib(n-1) + Fib(n-2)
month 1 2 3 4 5 6 7
rabbits 1 1 3 5
2 8 13

23. Sequences That Sum To n

24. Sequences That Sum To n
Let fn+1 be the number of different
sequences of 1’s and 2’s that sum to n.

25. Sequences That Sum To n
Let fn+1 be the number of different
sequences of 1’s and 2’s that sum to n.
f1 = 1

26. Sequences That Sum To n
Let fn+1 be the number of different
sequences of 1’s and 2’s that sum to n.
f1 = 1 0 = the empty sum

27. Sequences That Sum To n
Let fn+1 be the number of different
sequences of 1’s and 2’s that sum to n.
f1 = 1 0 = the empty sum
f2 = 1

28. Sequences That Sum To n
Let fn+1 be the number of different
sequences of 1’s and 2’s that sum to n.
f1 = 1 0 = the empty sum
f2 = 1 1 = 1

29. Sequences That Sum To n
Let fn+1 be the number of different
sequences of 1’s and 2’s that sum to n.
f1 = 1 0 = the empty sum
f2 = 1 1 = 1
f3 = 2

30. Sequences That Sum To n
2 = 1 + 1
2
Let fn+1 be the number of different
sequences of 1’s and 2’s that sum to n.
f1 = 1 0 = the empty sum
f2 = 1 1 = 1
f3 = 2

31. Sequences That Sum To n
Let fn+1 be the number of different
sequences of 1’s and 2’s that sum to n.
4 =

32. 2 + 2
2 + 1 + 1
1 + 2 + 1
1 + 1 + 2
1 + 1 + 1 + 1
Sequences That Sum To n
Let fn+1 be the number of different
sequences of 1’s and 2’s that sum to n.
4 =

33. Sequences That Sum To n
Let fn+1 be the number of different
sequences of 1’s and 2’s that sum to n.

34. fn+1 = fn + fn-1
Sequences That Sum To n
Let fn+1 be the number of different
sequences of 1’s and 2’s that sum to n.

35. fn+1 = fn + fn-1
# of
sequences
beginning
with a 2
# of
sequences
beginning
with a 1
Sequences That Sum To n
Let fn+1 be the number of different
sequences of 1’s and 2’s that sum to n.

36. Fibonacci Numbers Again
fn+1 = fn + fn-1
f1 = 1 f2 = 1
Let fn+1 be the number of different
sequences of 1’s and 2’s that sum to n.

37. Visual Representation: Tiling
Let fn+1 be the number of different
ways to tile a 1 × n strip with squares
and dominoes.

38. 1 way to tile a strip of length 0
1 way to tile a strip of length 1:
2 ways to tile a strip of length 2:
Visual Representation: Tiling

39. fn+1 = fn + fn-1
fn+1 is number of ways to tile length n.
fn tilings that start with a square.
fn-1 tilings that start with a domino.

40. Fibonacci Identities

41. Fibonacci Identities
Some examples:

42. Fibonacci Identities
Some examples:

43. Fibonacci Identities
Some examples:
F2n = F1 + F3 + F5 + … + F2n-1

44. Fibonacci Identities
Some examples:
F2n = F1 + F3 + F5 + … + F2n-1

45. Fibonacci Identities
Some examples:
F2n = F1 + F3 + F5 + … + F2n-1
Fm+n+1 = Fm+1 Fn+1 + Fm Fn

46. Fibonacci Identities
Some examples:
F2n = F1 + F3 + F5 + … + F2n-1
Fm+n+1 = Fm+1 Fn+1 + Fm Fn

47. Fibonacci Identities
Some examples:
F2n = F1 + F3 + F5 + … + F2n-1
Fm+n+1 = Fm+1 Fn+1 + Fm Fn
(Fn)2 = Fn-1 Fn+1 + (-1)n

48. Fm+n+1 = Fm+1 Fn+1 + Fm Fn

49. Fm+n+1 = Fm+1 Fn+1 + Fm Fn
m n

50. Fm+n+1 = Fm+1 Fn+1 + Fm Fn
m n
m-1 n-1

51. (Fn)2 = Fn-1 Fn+1 + (-1)n

52. (Fn)2 = Fn-1 Fn+1 + (-1)n
n-1
Fn tilings of a strip of length n-1

53. (Fn)2 = Fn-1 Fn+1 + (-1)n
n
(Fn)2 tilings of two strips of size n-1

54. (Fn)2 = Fn-1 Fn+1 + (-1)n
n
Draw a vertical “fault
line” at the rightmost
position (<n) possible
without cutting any
dominoes

55. (Fn)2 = Fn-1 Fn+1 + (-1)n
n
Swap the tails at the fault
line to map to a tiling of 2
(n-1)’s to a tiling of an
n-2 and an n.

56. (Fn)2 = Fn-1 Fn+1 + (-1)n-1
n even
n odd

57. Sneezwort (Achilleaptarmica)
Each time the plant starts a new shoot
it takes two months before it is strong
enough to support branching.

58. Counting Petals

59. Counting Petals
5 petals: buttercup, wild rose, larkspur,
columbine (aquilegia)
8 petals: delphiniums
13 petals: ragwort, corn marigold,
cineraria,
some daisies
21 petals: aster, black-eyed susan, chicory
34 petals: plantain, pyrethrum
55, 89 petals: michaelmas daisies, the
asteraceae family.

60. The Fibonacci Quarterly

61. Definition of φ (Euclid)

62. Ratio obtained when you divide a line segment
into two unequal parts such that the ratio of
the whole to the larger part is the same as the
ratio of the larger to the smaller.
Definition of φ (Euclid)

63. Ratio obtained when you divide a line segment
into two unequal parts such that the ratio of
the whole to the larger part is the same as the
ratio of the larger to the smaller.
Definition of φ (Euclid)
A B C

64. Ratio obtained when you divide a line segment
into two unequal parts such that the ratio of
the whole to the larger part is the same as the
ratio of the larger to the smaller.
Definition of φ (Euclid)
A B C
φ =
AC
AB
AB
BC
=

65. Ratio obtained when you divide a line segment
into two unequal parts such that the ratio of
the whole to the larger part is the same as the
ratio of the larger to the smaller.
Definition of φ (Euclid)
A B C
φ =
AC
AB
AB
BC
=
φ2 =

66. Ratio obtained when you divide a line segment
into two unequal parts such that the ratio of
the whole to the larger part is the same as the
ratio of the larger to the smaller.
Definition of φ (Euclid)
A B C
φ =
AC
AB
AB
BC
=
φ2 =
AC
BC

67. Ratio obtained when you divide a line segment
into two unequal parts such that the ratio of
the whole to the larger part is the same as the
ratio of the larger to the smaller.
Definition of φ (Euclid)
A B C
φ =
AC
AB
AB
BC
=
φ2 =
AC
BC
φ 2 - φ =

68. Ratio obtained when you divide a line segment
into two unequal parts such that the ratio of
the whole to the larger part is the same as the
ratio of the larger to the smaller.
Definition of φ (Euclid)
A B C
φ =
AC
AB
AB
BC
=
φ2 =
AC
BC
φ 2 - φ =
AC
BC
AB
BC
- =

69. Ratio obtained when you divide a line segment
into two unequal parts such that the ratio of
the whole to the larger part is the same as the
ratio of the larger to the smaller.
Definition of φ (Euclid)
A B C
φ =
AC
AB
AB
BC
=
φ2 =
AC
BC
φ 2 - φ =
AC
BC
AB
BC
BC
BC
- = = 1

70. φ2 – φ – 1 = 0

71. φ2 – φ – 1 = 0
φ =
1 + √5
2

73. Golden ratio supposed to arise
in…
Parthenon, Athens (400 B.C.)
b
a
The great pyramid at Gizeh
Ratio of a person’s height
to the height of his/her navel

74. Golden ratio supposed to arise
in…
Parthenon, Athens (400 B.C.)
b
a
The great pyramid at Gizeh
Ratio of a person’s height
to the height of his/her navel
Mostly
circumstantial
evidence…

75. Expanding Recursively

76. Expanding Recursively

77. Continued Fraction
Representation

78. A (Simple) Continued Fraction Is Any
Expression Of The Form:
where a, b, c, … are whole numbers.

79. A Continued Fraction can have a finite
or infinite number of terms.
We also denote this fraction by [a,b,c,d,e,f,…]

80. A Finite Continued Fraction
Denoted by [2,3,4,2,0,0,0,…]

81. An Infinite Continued Fraction
Denoted by [1,2,2,2,…]

82. Recursively Defined Form For CF

83. Continued fraction representation of a
standard fraction

84. e.g., 67/29 = 2 with remainder 9/29
= 2 + 1/ (29/9)

85. Ancient Greek Representation:
Continued Fraction Representation

86. Ancient Greek Representation:
Continued Fraction Representation
= [1,1,1,1,0,0,0,…]

87. Ancient Greek Representation:
Continued Fraction Representation

88. Ancient Greek Representation:
Continued Fraction Representation
= [1,1,1,1,1,0,0,0,…]

89. Ancient Greek Representation:
Continued Fraction Representation
= [1,1,1,1,1,1,0,0,0,…]

90. A Pattern?
Let r1 = [1,0,0,0,…] = 1
r2 = [1,1,0,0,0,…] = 2/1
r3 = [1,1,1,0,0,0…] = 3/2
r4 = [1,1,1,1,0,0,0…] = 5/3
and so on.
Theorem:
rn = Fib(n+1)/Fib(n)

91. 1,1,2,3,5,8,13,21,34,55,….

92. 1,1,2,3,5,8,13,21,34,55,….
2/1 = 2

93. 1,1,2,3,5,8,13,21,34,55,….
2/1 = 2
3/2 = 1.5

94. 1,1,2,3,5,8,13,21,34,55,….
2/1 = 2
3/2 = 1.5
5/3 = 1.666…

95. 1,1,2,3,5,8,13,21,34,55,….
2/1 = 2
3/2 = 1.5
5/3 = 1.666…
8/5 = 1.6

96. 1,1,2,3,5,8,13,21,34,55,….
2/1 = 2
3/2 = 1.5
5/3 = 1.666…
8/5 = 1.6
13/8 = 1.625

97. 1,1,2,3,5,8,13,21,34,55,….
2/1 = 2
3/2 = 1.5
5/3 = 1.666…
8/5 = 1.6
13/8 = 1.625
21/13 = 1.6153846…

98. 1,1,2,3,5,8,13,21,34,55,….
2/1 = 2
3/2 = 1.5
5/3 = 1.666…
8/5 = 1.6
13/8 = 1.625
21/13 = 1.6153846…
34/21 = 1.61904…

99. 1,1,2,3,5,8,13,21,34,55,….
2/1 = 2
3/2 = 1.5
5/3 = 1.666…
8/5 = 1.6
13/8 = 1.625
21/13 = 1.6153846…
34/21 = 1.61904…
φ = 1.6180339887498948482045

100. Pineapple whorls
Church and Turing were both
interested in the number of
whorls in each ring of the
spiral.
The ratio of consecutive ring
lengths approaches the
Golden Ratio.

102. Proposition:
Any finite continued fraction
evaluates to a rational.
Theorem
Any rational has a finite
continued fraction
representation.

103. Hmm.
Finite CFs = Rationals.
Then what do
infinite continued fractions
represent?

104. An infinite continued fraction

105. Quadratic Equations
• X2 – 3x – 1 = 0
• X2 = 3X + 1
• X = 3 + 1/X
• X = 3 + 1/X = 3 + 1/[3 + 1/X] = …

106. A Periodic CF

107. Theorem:
Any solution to a quadratic
equation has a periodic
continued fraction.
Converse:
Any periodic continued
fraction is the solution of a
quadratic equation.
(try to prove this!)

108. So they express more
than just the rationals…
What about those
non-recurring infinite
continued fractions?

109. Non-periodic CFs

110. What is the pattern?

111. What is the pattern?
No one knows!

112. What a cool representation!
Finite CF: Rationals
Periodic CF: Quadratic roots
And some numbers reveal
hidden regularity.

113. More good news: Convergents
Let α = [a1, a2, a3, ...] be a CF.
Define: C1 = [a1,0,0,0,0..]
C2 = [a1,a2,0,0,0,...]
C3 = [a1,a2,a3,0,0,...] and so on.
Ck is called the k-th convergent of α
α is the limit of the sequence C1, C2, C3,…

114. Best Approximator Theorem
• A rational p/q is the best approximator to
a real α if no rational number of
denominator smaller than q comes closer
to α.

115. Best Approximator Theorem
• A rational p/q is the best approximator to
a real α if no rational number of
denominator smaller than q comes closer
to α.
BEST APPROXIMATOR THEOREM:
Given any CF representation of α,
each convergent of the CF is a
best approximator for α !

116. Best Approximators of π
C1 = 3
C2 = 22/7
C3 = 333/106
C4 = 355/113
C5 = 103993/33102
C6 =104348/33215

117. Continued Fraction
Representation

118. Continued Fraction
Representation

119. Remember?
We already saw the convergents of this CF
[1,1,1,1,1,1,1,1,1,1,1,…]
are of the form Fib(n+1)/Fib(n)
Hence:

120. 1,1,2,3,5,8,13,21,34,55,….
• 2/1 = 2
• 3/2 = 1.5
• 5/3 = 1.666…
• 8/5 = 1.6
• 13/8 = 1.625
• 21/13 = 1.6153846…
• 34/21 = 1.61904…
• φ = 1.6180339887498948482045...

121. As we’ve seen...

122. Going the Other Way

123. What is the Power Series
Expansion of z/(1-z-z2) ?
What does this look like
when we expand it as an
infinite sum?

124. Since the bottom is quadratic we
can factor it:

125. Write as a sum of two terms
where
Solving:

126. Now use the geometric series

128. The nth Fibonacci number is:
Leonhard Euler (1765)
J. P. M. Binet (1843)
A de Moivre (1730)

131. 74
Recurrences and generating
functions
Golden ratio
Continued fractions
Convergents
Closed form for Fibonaccis
Here’s What
You Need to
Know…