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Recurrences and Continued Fractions

Recurrences and Continued Fractions. Solve in integers x 1 + x 2 +…+ x 5 = 40 x k r k; . y k =x k – k y 1 + y 2 +…+ y 5 = 25 y k ≥ 0 ; . Solve in integers x + y + z = 11 x r 0; 0  y  3; z r 0. [X 11 ]. Partitions . Find the number of ways to partition the integer n

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Recurrences and Continued Fractions

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  1. Recurrences and Continued Fractions

  2. Solve in integers x1 + x2 +…+ x5 = 40 xkrk; yk=xk – k y1 + y2 +…+ y5 = 25 yk≥ 0;

  3. Solve in integers x + y + z = 11 xr0; 0y3; zr0 [X11]

  4. 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

  5. Plan • Review: Sat, 6-8pm in 2315 DH • Exam: Mon in recitations • Characteristic Equations • Golden Ratio • Continued Fractions Applied Combinatorics, by Alan Tucker The Divine Proportion, by H. E. Huntley

  6. Leonardo Fibonacci In 1202, Fibonacci has become interested in rabbits and what they do

  7. 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

  8. F0=0, F1=1, Fn=Fn-1+Fn-2 for n≥2 What is a closed form formula for Fn? We won’t be using GFs!

  9. WARNING!!!! This lecture has explicit mathematical content that can be shocking to some students.

  10. Characteristic Equation Fn = Fn-1 + Fn-2 Consider solutions of the form: Fn= n for some (unknown) constant  ≠ 0  must satisfy: n - n-1 - n-2 = 0

  11. n - n-1 - n-2 = 0 Characteristic equation iffn-2(2- - 1) = 0 iff2- - 1= 0  = , or  = -1/  (“phi”) is the golden ratio

  12. a,ba n + b (-1/)nsatisfies the inductive condition So for all these values of the inductive condition is satisfied: 2- - 1= 0 Do any of them happen to satisfy the base condition as well? F0=0, F1=1

  13. a,ba 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

  14. Leonhard Euler (1765)

  15. Fibonacci Power Series

  16. Fibonacci Bamboozlement 8 13 5 8

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

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

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

  20. From the previous lecture

  21. Characteristic Equation a, b

  22. Characteristic Equation

  23. Characteristic Equation Theorem. Let  be a root of multiplicity p of the characteristic equation. Then are all solutions.

  24. The theorem says that xn=n is a solution. This can be easily verified: n = 2 n - n

  25. From the previous lecture: Rogue Recurrence Characteristic equation:

  26. General solution:

  27. The Golden Ratio “Some other majors have their mystery numbers, like  and e. In Computer Science the mystery number is , the Golden Ratio.” – S.Rudich

  28. 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 C B

  29. Golden Ratio - the divine proportion  = 1.6180339887498948482045… “Phi” is named after the Greek sculptor Phidias

  30. 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).

  31. Which is the most attractive rectangle?

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

  33. The Golden Ratio

  34. DivinaProportioneLuca 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

  35. 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

  36. Table of contents “For the sake of salvation, the list must end here” – Luca Pacioli

  37. DivinaProportioneLuca Pacioli (1509) "Ninth Most Excellent Effect" two diagonals of a regular pentagon divide each other in the Divine Proportion. C A B

  38. Expanding Recursively

  39. Expanding Recursively

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

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

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

  43. Recursively Defined Form For CF

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

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

  46. Euclid’s GCD = Continued Fraction

  47. Euclid’s GCD = Continued Fraction

  48. 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:  = Fn/Fn-1 when n ->

  49. Divine Proportion

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