Recurrences, Phi and CF

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Recurrences, Phi and CF
Great Theoretical Ideas In Computer Science Great Theoretical Ideas In Computer Science Great Theoretical Ideas In Computer Science
S. Rudich V. Adamchik CS 15-251 Spring 2006
Lecture 13 Feb 28, 2006 Carnegie Mellon University
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Warm-up
Solve in integers x1x2x311 x1r0 x2b3 x3r0
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Warm-up
Solve in integers x1x2x311 x1r0 x2b3 x3r0
X11
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Make Change (interview question)
Find the number of ways to make change for 1
using pennies, nickels, dimes and quarters
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Make Change
Find the number of ways to make change for 1
using pennies, nickels, dimes and
quarters X100
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Make Change
Find the number of ways to make change for 1
using pennies, nickels, dimes and quarters
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Partitions
Find the number of ways to partition the integer
n 3 11112 411111121322
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Partitions
Find the number of ways to partition the integer
n
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Partitions
Find the number of ways to partition the integer
n
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Leonardo Fibonacci

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

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The rabbit reproduction model

• A rabbit lives forever
• The population starts as a 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

month 1 2 3 4 5 6 7
rabbits 1 1 2 3 5 8 13
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Fn is called the nth Fibonacci number
month 1 2 3 4 5 6 7
rabbits 1 1 2 3 5 8 13
F00, F11, and FnFn-1Fn-2 for n?2
Fn is defined by a recurrence relation.
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F00, F11, FnFn-1Fn-2 for n?2
What is a closed form formula for Fn?
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Techniques you have seen so far
FnFn-1Fn-2 Consider
solutions of the form Fn cn
for some constant c c must satisfy cn -
cn-1 - cn-2 0
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cn - cn-1 - cn-2 0

• iff cn-2(c2 - c - 1) 0
• iff c0 or c2 - c - 1 0
• Iff c 0, c ?, or c -1/?
• ? (phi) is the golden ratio

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c 0, c ?, or c -(1/?)

• So for all these values of c the inductive
  condition is satisfied
• cn - cn-1 - cn-2 0
• Do any of them happen to satisfy the base
  condition as well?
• F00, F11

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?a,b a ?n b (-1/ ?)n satisfies the inductive
condition

• Adjust a and b to fit the base conditions.
• n0 ab 0
• n1 a ?1 b (-1/ ?)1 1
• a 1/?5 b -1/?5

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Leonhard Euler (1765)
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Fibonacci Power Series
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Fibonacci Bamboozlement
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Cassinis Identity
Fn1Fn-1 - Fn2 (-1)n We
dissect FnxFn square and rearrange pieces into
Fn1xFn-1 square
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Golden Ratio Divine Proportion

• 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.

A
B
C
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Ratio of height of the person to height of a
persons navel
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Aesthetics

• ? plays a central role in renaissance art and
  architecture.
• After measuring the dimensions of pictures,
  cards, books, snuff boxes, writing paper,
  windows, and such, psychologist Gustav Fechner
  claimed that the preferred rectangle had sides in
  the golden ratio (1871).

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Which is the most attractive rectangle?
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Which is the most attractive rectangle?
?
Golden Rectangle
1
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Divina ProportioneLuca Pacioli (1509)

• Pacioli devoted an entire book to the marvelous
  properties of ?. The book was illustrated by a
  friend of his named

Leonardo Da Vinci
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Table of contents

• The first considerable effect
• The second essential effect
• The third singular effect
• The fourth ineffable effect
• The fifth admirable effect
• The sixth inexpressible effect
• The seventh inestimable effect
• The ninth most excellent effect
• The twelfth incomparable effect
• The thirteenth most distinguished effect

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Divina ProportioneLuca Pacioli (1509)

• "Ninth Most Excellent Effect"
• two diagonals of a regular pentagon divide each
  other in the Divine Proportion.

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Expanding Recursively
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Expanding Recursively
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Expanding Recursively
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Expanding Recursively
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A (Simple) Continued Fraction Is Any Expression
Of The Form
where a, b, c, are whole numbers.
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A Continued Fraction can have a finite or
infinite number of terms.
We also denote this fraction by a,b,c,d,e,f,
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Continued Fraction Representation
1,1,1,1,1,0,0,0,
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Recursively Defined Form For CF
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Proposition Any finite continued fraction
evaluates to a rational. Converse Any rational
has a finite continued fraction representation.
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Euclids GCD Continued Fractions
Euclid(A,B) Euclid(B, A mod B) Stop when B0
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A Pattern for ?
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 Fn1/Fn
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Divine Proportion
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Heads-on
How to convert kilometers into miles?
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Magic conversion
50 km 34 13 3 50 F9 F7 F4 F8 F6
F3 31 miles
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Quadratic Equations

• X2 3x 1 0
• X2 3X 1
• X 3 1/X
• X 3 1/X 3 1/3 1/X

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A Periodic CF
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A period-2 CF
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Proposition Any quadratic solution has a
periodic continued fraction. Converse Any
periodic continued fraction is the solution of a
quadratic equation
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What about those non-periodic continued
fractions?
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Non-periodic CFs
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What is the pattern?
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What a cool representation! Finite CF
Rationals Periodic CF Quadratic roots And some
numbers reveal hidden regularity.
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More recurrences!! Let us embark now on the
Catalan numbers
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Counting Binary Trees
Count the number of binary trees with n nodes.
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Counting Binary Trees
Let Tn represent the number of trees with n
nodes T11, T22
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Counting Binary Trees
Size n

Size n-1-k
Size k
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Counting Binary Trees
Size n

Size n-1-k
Size k

• Tn T0 Tn-1 T1 Tn-2Tn-1 T0

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Counting Binary Trees
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Triangulation
Count the number of ways to divide a convex
n-gon into triangles with noncrossing
diagonals.
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• Review GCD algorithm
• Recurrences, Phi and CF
• The Catalan numbers

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