The Fibonacci Numbers And An Unexpected CalculationLeonardo FibonacciInductive Definition or Recurrence Relation for the Fibonacci NumbersSneezwort (Achilleaptarmica)Counting PetalsPineapple whorlsSlide 7Bernoulli Spiral When the growth of the organism is proportional to its sizeSlide 9Slide 10Slide 11Golden Ratio: the divine proportionDefinition of (Euclid)Expanding RecursivelyContinued Fraction RepresentationSlide 16Remember?Slide 18Slide 19Sequences That Sum To nSlide 21Slide 22Slide 23Slide 24Slide 25Fibonacci Numbers AgainVisual Representation: TilingSlide 28Slide 29fn+1 = fn + fn-1Slide 31Fibonacci IdentitiesFm+n+1 = Fm+1 Fn+1 + Fm Fn(Fn)2 = Fn-1 Fn+1 + (-1)nSlide 35Slide 36Slide 37Slide 38Slide 39Slide 40(Fn)2 = Fn-1 Fn+1 + (-1)n-1More random factsThe Fibonacci QuarterlySlide 44How to divide polynomials?The Geometric SeriesSlide 47The Infinite Geometric SeriesSlide 49Slide 50Something a bit more complicatedHenceGoing the Other WaySlide 54ThusSlide 56Slide 57Formal Power SeriesAddition and MultiplicationMultiplying two power seriesGeometric Series (Quadratic Form)Fibonacci NumbersGetting the Fibonacci Power SeriesSlide 64Slide 65Slide 66Linear factors on the bottomSlide 68Slide 69Power Series Expansion of FSlide 71Slide 72Slide 73Slide 74Slide 75Slide 76Slide 77Slide 78Slide 79Some simple power seriesAl-Karaji’s IdentitiesSlide 82Slide 83Slide 84Slide 85Slide 86Coefficient of Xk in PV = (X2+X)(1-X)-4 is the sum of the first k squares:Polynomials give us closed form expressionsREFERENCESStudy BeeThe Fibonacci Numbers And An Unexpected CalculationGreat Theoretical Ideas In Computer ScienceAnupam GuptaCS 15-251 Fall 2005Lecture 13 October 11, 2005 Carnegie Mellon UniversityLeonardo FibonacciIn 1202, Fibonacci proposed a problem about the growth of rabbit populations.Inductive Definition or Recurrence Relation for theFibonacci Numbers Stage 0, Initial Condition, or Base Case:Fib(0) = 0; Fib (1) = 1Inductive RuleFor n>1, Fib(n) = Fib(n-1) + Fib(n-2)n 0 1 2 3 4 5 6 7Fib(n) 0 1 1 2 3 5 813Sneezwort (Achilleaptarmica) Each time the plant starts a new shoot it takes two months before it is strong enough to support branching.Counting Petals5 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, theasteraceae family.Pineapple whorlsChurch 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.Bernoulli Spiral When the growth of the organism is proportional to its sizeBernoulli Spiral When the growth of the organism is proportional to its sizeIs there life after andeGolden Ratio: the divine proportion = 1.6180339887498948482045…“Phi” is named after the Greek sculptor Phidias ACABABBCACBCACBCABBCBCBC11 0222Definition 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 CExpanding Recursively 111111111111Continued Fraction Representation 1111111111111111111 ....Continued Fraction Representation1111111111111111111 ....1 52= +++++++++++RememberWe already saw the convergents of this CF[1,1,1,1,1,1,1,1,1,1,1,…]are of the formFib(n+1)/Fib(n)n11 5l m2i��-+=f =nnFFHence:Continued Fraction Representation 1111111111111111111 ....Let us take a slight detour and look at a different representation.Sequences That Sum To nLet fn+1 be the number of different sequences of 1’s and 2’s that sum to n.Example: f5 = 5Sequences That Sum To nLet fn+1 be the number of different sequences of 1’s and 2’s that sum to n.Example: f5 = 54 = 2 + 22 + 1 + 11 + 2 + 11 + 1 + 21 + 1 + 1 + 1Sequences That Sum To nf1f2f3Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.Sequences That Sum To nf1 = 1 0 = the empty sumf2 = 1 1 = 1f3 = 2 2 = 1 + 1 2Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.Sequences That Sum To nfn+1 = fn + fn-1Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.Sequences That Sum To nfn+1 = fn + fn-1Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.# of sequences beginning with a 2# of sequences beginning with a 1Fibonacci Numbers Againffn+1n+1 = f = fnn + f + fn-1n-1ff11 = 1 f = 1 f22 = 1 = 1Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.Visual Representation: TilingLet fn+1 be the number of different ways to tile a 1 × n strip with squares and dominoes.Visual Representation: TilingLet fn+1 be the number of different ways to tile a 1 × n strip with squares and dominoes.Visual Representation: Tiling1 way to tile a strip of length 01 way to tile a strip of length 1:2 ways to tile a strip of length 2:fn+1 = fn + fn-1fn+1 is number of ways to title length n.fn tilings that start with a square.fn-1 tilings that start with a domino.Let’s use this visual representation to prove a couple of Fibonacci identities.Fibonacci IdentitiesSome examples:F2n = F1 + F3 + F5 + … + F2n-1Fm+n+1 = Fm+1 Fn+1 + Fm Fn(Fn)2 = Fn-1 Fn+1 + (-1)nFm+n+1 = Fm+1 Fn+1 + Fm Fnmmnnm-1m-1n-1n-1(Fn)2 = Fn-1 Fn+1 + (-1)n(Fn)2 = Fn-1 Fn+1 + (-1)nn-1n-1Fn tilings of a strip of length n-1(Fn)2 = Fn-1 Fn+1 + (-1)nn-1n-1n-1n-1(Fn)2 = Fn-1 Fn+1 + (-1)nnn(Fn)2 tilings of two strips of size n-1(Fn)2 = Fn-1 Fn+1 + (-1)nnnDraw a vertical “fault Draw a vertical “fault line” at the line” at the rightmost rightmost position position (<n)(<n) possible without possible without cutting any cutting any dominoes dominoes(Fn)2 = Fn-1 Fn+1 + (-1)nnnSwap the tailsSwap the tails at the at the fault linefault line to map to a to map to a tiling of 2 n-1 ‘s to a tiling of 2 n-1 ‘s to a tiling of an n-2 and an tiling of an n-2 and an n.n.(Fn)2 = Fn-1 Fn+1 + (-1)nnnSwap the tailsSwap the tails at the at the fault linefault line to map to a to map to a tiling of 2 n-1 ‘s to a tiling of 2 n-1 ‘s to a tiling of an n-2 and an
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