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Fibonacci Numbers, Polynomial Coefficients, and Vector Programs.

Explore the fascinating world of Fibonacci numbers, polynomial coefficients, and vector programs. Learn about the growth of rabbit populations, counting petals, geometric series, and more. Unlock the secrets of mathematical patterns and applications.

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Fibonacci Numbers, Polynomial Coefficients, and Vector Programs.

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  1. Fibonacci Numbers, Polynomial Coefficients, and Vector Programs.

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

  3. Inductive Definition or Recurrence Relation for theFibonacci Numbers Stage 0, Initial Condition, or Base Case: Fib(0) = 0; Fib (1) = 1 Inductive Rule For n>1, Fib(n) = Fib(n-1) + Fib(n-2)

  4. Sneezwort (Achilleaptarmica) 0 Each time the plant starts a new shoot it takes two months before it is strong enough to support branching. See http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

  5. 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.

  6. 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.

  7. coneflower

  8. Definition of  (Euclid) • 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

  9. Expanding Recursively

  10. Continued Fraction Representation

  11. Continued Fraction Representation

  12. 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… •  = 1.6180339887498948482045

  13. 1 + X + X2 1 – X 1 -(1 – X) X2 X3 X -(X – X2) -(X2 – X3) 1 + X + X2 + X3 + X4+ X5 + X6 + X7 + … = How to divide polynomials? 1 ? 1 – X …

  14. 1 1 + X1 + X2 + X3 + … + Xn + ….. = 1 - X The Infinite Geometric Series

  15. 1 1 + X1 + X2 + X3 + … + Xn + ….. = 1 - X • (1-X) ( 1 + X1 + X2 + X 3 + … + Xn + … ) • = 1 + X1 + X2 + X 3 + … + Xn + Xn+1 + …. • - X1 - X2 - X 3 - … - Xn-1 – Xn - Xn+1 - … • = 1

  16. 1 1 + X1 + X2 + X3 + … + Xn + ….. = 1 - X 1 + X 1 – X 1 -(1 – X) X2 X3 X -(X – X2) + X2 + … -(X2 – X3) …

  17. X + X2 + 2X3 + 3X4 + 5X5 + 8X6 1 – X – X2 X -(X – X2 – X3) X2 + X3 -(X2 – X3 – X4) 2X3 + X4 -(2X3 – 2X4 – 2X5) 3X4 + 2X5 -(3X4 – 3X5 – 3X6) 5X5 + 3X6 8X6 + 5X7 -(5X5 – 5X6 – 5X7) -(8X6 – 8X7 – 8X8) Something a bit more complicated X 1 – X – X2

  18. Hence X • = F0 1 + F1 X1 + F2 X2 +F3 X3 + F4 X4 + F5 X5 + F6 X6 + … 1 – X – X2 = 01 + 1 X1 + 1 X2 + 2X3 + 3X4 + 5X5 + 8X6 + …

  19. Going the Other Way F0 = 0, F1 = 1 • (1 - X- X2)  ( F0 1 + F1 X1 + F2 X2 + … + Fn-2 Xn-2 + Fn-1 Xn-1 + Fn Xn + …

  20. Going the Other Way • (1 - X- X2)  ( F0 1 + F1 X1 + F2 X2 + … + Fn-2 Xn-2 + Fn-1 Xn-1 + Fn Xn + … • = ( F0 1 + F1 X1 + F2 X2 + … + Fn-2 Xn-2 + Fn-1 Xn-1 + Fn Xn + … • - F0 X1 - F1 X2 - … - Fn-3 Xn-2 - Fn-2 Xn-1 - Fn-1 Xn - … • - F0 X2 - … - Fn-4 Xn-2 - Fn-3 Xn-1 - Fn-2 Xn - … • = F0 1 + ( F1 – F0 ) X1 F0 = 0, F1 = 1 = X

  21. Thus • F0 1 + F1 X1 + F2 X2 + … + Fn-1 Xn-1 + Fn Xn + … X = 1 – X – X2 = X/(1- X)(1 – (-)-1 X)  (--1)= -1, -+-1= -1

  22. Linear factors on the bottom X (1 – X)(1- (-)-1X) ? n=0..∞ Xn =

  23. an+1 – bn+1 a- b Geometric Series (Quadratic Form) (1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn + …..) = 1 = (1 – aX)(1-bX) n=0..∞ Xn =

  24. n+1 – (--1)n+1 √5 Geometric Series (Quadratic Form) 1 (1 – X)(1- (--1X)) n=0.. ∞ Xn =

  25. n+1 – (--1)n+1 √5 Power Series Expansion of F X (1 – X)(1- (--1X) n=0.. ∞ Xn+1 =

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

  27. an+1 – bn+1 a- b Let’s Derive This (1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn + …..) = 1 = (1 – aX)(1-bX) n=0..∞ Xn =

  28. Substituting Y = aX … 1 1 + Y1 + Y2 + Y 3 + … + Yn + ….. = 1 - Y

  29. Geometric Series (Linear Form) 1 1 + aX1 + a2X2 + a3X 3 + … + anXn + ….. = 1 - aX

  30. Geometric Series (Quadratic Form) (1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn + …..) = 1 (1 – aX)(1-bX)

  31. Suppose we multiply this out to get a single, infinite polynomial.What is an expression for Cn? (1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn + …..) = 1 + c1X1 + .. + ck Xk + …

  32. cn =a0bn + a1bn-1 +… aibn-i… + an-1b1 + anb0 (1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn + …..) = 1 + c1X1 + .. + ck Xk + …

  33. If a = b then cn = (n+1)(an)a0bn + a1bn-1 +… aibn-i… + an-1b1 + anb0 (1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn + …..) = 1 + c1X1 + .. + ck Xk + …

  34. an+1 – bn+1 a- b a0bn + a1bn-1 +… aibn-i… + an-1b1 + anb0 = • (a-b) (a0bn + a1bn-1 +… aibn-i… + an-1b1 + anb0) • = a1bn +… ai+1bn-i… + anb1 + an+1b0 • - a0bn+1 – a1bn… ai+1bn-i… - an-1b2 - anb1 • = - bn+1 + an+1 • = an+1 – bn+1

  35. an+1 – bn+1 a- b if a  b then cn =a0bn + a1bn-1 +… aibn-i… + an-1b1 + anb0 (1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn + …..) = 1 + c1X1 + .. + ck Xk + …

  36. an+1 – bn+1 a- b Geometric Series (Quadratic Form) (1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn + …..) = 1 = (1 – aX)(1-bX) n=0.. Xn = n=0.. or Xn (n+1)an when a=b

  37. Sequences That Sum To n • Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n. • Example: f5 = 5

  38. Sequences That Sum To n • Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n. • Example: f5 = 5 • 4 = 2 + 2 • 2 + 1 + 1 • 1 + 2 + 1 • 1 + 1 + 2 • 1 + 1 + 1 + 1

  39. 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 f3 f2

  40. 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 f3 = 2 2 = 1 + 1 2 f2 = 1 1 = 1

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

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

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

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

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

  46. Visual Representation: Tiling • 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:

  47. 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.

  48. Let’s use this visual representation to prove a couple of Fibonacci identities.

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