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Recurrences, Phi and CF

Recurrences, Phi and CF. Warm-up. Solve in integers x 1 +x 2 +x 3 =11 x 1 r 0; x 2 b 3; x 3 r 0. Warm-up. Solve in integers x 1 +x 2 +x 3 =11 x 1 r 0; x 2 b 3; x 3 r 0 [X 11 ]. Make Change (interview question).

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Recurrences, Phi and CF

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  1. Recurrences, Phi and CF

  2. Warm-up Solve in integers x1+x2+x3=11 x1r0; x2b3; x3r0

  3. Warm-up Solve in integers x1+x2+x3=11 x1r0; x2b3; x3r0 [X11]

  4. Make Change (interview question) Find the number of ways to make change for $1 using pennies, nickels, dimes and quarters

  5. Make Change Find the number of ways to make change for $1 using pennies, nickels, dimes and quarters [X100]

  6. Make Change Find the number of ways to make change for $1 using pennies, nickels, dimes and quarters

  7. Partitions Find the number of ways to partition the integer n 3 = 1+1+1=1+2 4=1+1+1+1=1+1+2=1+3=2+2

  8. Partitions Find the number of ways to partition the integer n

  9. Partitions Find the number of ways to partition the integer n

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

  11. 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

  12. Fn is called the nth Fibonacci number F0=0, F1=1, and Fn=Fn-1+Fn-2 for n2 Fn is defined by a recurrence relation.

  13. F0=0, F1=1, Fn=Fn-1+Fn-2 for n2 What is a closed form formula for Fn?

  14. Techniques you have seen so far • Fn=Fn-1+Fn-2 • Consider solutions of the form: • Fn= cn • for some constant c • c must satisfy: • cn - cn-1 - cn-2 = 0

  15. cn - cn-1 - cn-2 = 0 • iff cn-2(c2 - c- 1) = 0 • iff c=0 or c2 - c- 1= 0 • Iff c = 0, c = , or c = -1/ •  (“phi”) is the golden ratio

  16. 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? • F0=0, F1=1

  17. a,b a n + b (-1/ )nsatisfies the inductive condition • Adjust a and b to fit the base conditions. • n=0: a+b = 0 • n=1: a 1 + b (-1/ )1 = 1 • a= 1/5 b= -1/5

  18. Leonhard Euler (1765)

  19. Fibonacci Power Series

  20. Fibonacci Bamboozlement

  21. Cassini’s Identity • Fn+1Fn-1 - Fn2 = (-1)n • We dissect FnxFn square and rearrange pieces into Fn+1xFn-1 square

  22. 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

  23. Ratio of height of the person to height of a person’s navel

  24. 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).

  25. Which is the most attractive rectangle?

  26. Which is the most attractive rectangle?  Golden Rectangle 1

  27. 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

  28. 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

  29. Divina ProportioneLuca Pacioli (1509) • "Ninth Most Excellent Effect" • two diagonals of a regular pentagon divide each other in the Divine Proportion.

  30. Expanding Recursively

  31. Expanding Recursively

  32. Expanding Recursively

  33. Expanding Recursively

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

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

  36. Continued Fraction Representation = [1,1,1,1,1,0,0,0,…]

  37. Recursively Defined Form For CF

  38. Proposition: Any finite continued fraction evaluates to a rational. Converse: Any rational has a finite continued fraction representation.

  39. Euclid’s GCD = Continued Fractions Euclid(A,B) = Euclid(B, A mod B) Stop when B=0

  40. 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 = Fn+1/Fn

  41. Divine Proportion

  42. Heads-on How to convert kilometers into miles?

  43. Magic conversion 50 km = 34 + 13 + 3 50 = F9 + F7 + F4 F8 + F6 + F3 = 31miles

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

  45. A Periodic CF

  46. A period-2 CF

  47. Proposition: Any quadratic solution has a periodic continued fraction. Converse: Any periodic continued fraction is the solution of a quadratic equation

  48. What about those non-periodic continued fractions?

  49. Non-periodic CFs

  50. What is the pattern?

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