Fibonacci Numbers, Polynomial Coefficients, and Vector Programs.Leonardo FibonacciInductive Definition or Recurrence Relation for the Fibonacci NumbersSneezwort (Achilleaptarmica)Counting PetalsPineapple whorlsSlide 7Slide 8Definition of (Euclid)Expanding RecursivelyContinued Fraction RepresentationSlide 131,1,2,3,5,8,13,21,34,55,….How to divide polynomials?The Infinite Geometric SeriesSlide 17Slide 18Something a bit more complicatedHenceGoing the Other WaySlide 22ThusLinear factors on the bottomGeometric Series (Quadratic Form)Slide 26Power Series Expansion of FSlide 28Slide 29Let’s Derive ThisSubstituting Y = aX …Geometric Series (Linear Form)Geometric Series (Quadratic Form)Suppose we multiply this out to get a single, infinite polynomial. What is an expression for Cn?cn = a0bn + a1bn-1 +… aibn-i… + an-1b1 + anb0If a = b then cn = (n+1)(an) a0bn + a1bn-1 +… aibn-i… + an-1b1 + anb0Slide 37if a b then cn = a0bn + a1bn-1 +… aibn-i… + an-1b1 + anb0Slide 39Sequences That Sum To nSlide 41Slide 42Slide 43Slide 44Slide 45Fibonacci Numbers AgainVisual Representation: TilingSlide 48Slide 49fn+1 = fn + fn-1Slide 51Fibonacci IdentitiesFm+n+1 = Fm+1 Fn+1 + Fm Fn(Fn)2 = Fn-1 Fn+1 + (-1)nSlide 55Slide 56Slide 57Slide 58Slide 59Slide 60(Fn)2 = Fn-1 Fn+1 + (-1)n-1The Fibonacci QuarterlyVector ProgramsSlide 64Slide 65Slide 66Slide 67Slide 68Programs -----> PolynomialsFormal Power SeriesV! = < a0, a1, a2, . . . >Slide 72Slide 73Slide 74Slide 75What does this program output?Al Karaji Perfect SquaresSlide 78Slide 79Al-Karaji Program(1+X)/(1-X)3Slide 82Slide 83Slide 84Slide 85Slide 86Slide 87Slide 88Slide 89Vector programs -> Polynomials -> Closed form expressionREFERENCESFibonacci Numbers, Polynomial Coefficients, and Vector Programs.Leonardo 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 8 13Sneezwort (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. 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= +++++++++++1,1,2,3,5,8,13,21,34,55,….2/1 = 23/2 = 1.55/3 = 1.666…8/5 = 1.613/8 = 1.62521/13 = 1.6153846…34/21 = 1.61904… = 1.6180339887498948482045How to divide polynomials?1 1 1 – X?1 – X 11-(1 – X)X+ X-(X – X2)X2+ X2-(X2 – X3)X3=1 + X + X2 + X3 + X4 + X5 + X6 + X7 + … …1 + X1 + X11 + X + X22 + X + X33 + … + X + … + Xnn + ….. = + ….. = 11 1 - X1 - X The Infinite Geometric Series(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 1 + X1 + X11 + X + X22 + X + X33 + … + X + … + Xnn + ….. = + ….. = 11 1 - X1 - X1 + X1 + X11 + X + X22 + X + X33 + … + X + … + Xnn + ….. = + ….. = 11 1 - X1 - X 1 – X 11-(1 – X)X+ X-(X – X2)X2+ X2 + …-(X2 – X3)X3…X X 1 – X – X2Something a bit more complicated1 – X – X2XX2 + X3-(X – X2 – X3)X2X3 + X4-(X2 – X3 – X4)+ X2-(2X3 – 2X4 – 2X5)+ 2X33X4 + 2X5+ 3X4-(3X4 – 3X5 – 3X6)5X5 + 3X6+ 5X5-(5X5 – 5X6 – 5X7)8X6 + 5X7+ 8X6-(8X6 – 8X7 – 8X8)Hence= F0 1 + F1 X1 + F2 X2 +F3 X3 + F4 X4 + F5 X5 + F6 X6 + …X X 1 – X – X2= 01 + 1 X1 + 1 X2 + 2X3 + 3X4 + 5X5 + 8X6 + …Going the Other Way(1 - X- X2) ( F0 1 + F1 X1 + F2 X2 + … + Fn-2 Xn-2 + Fn-1 Xn-1 + Fn Xn + … F0 = 0, F1 = 1Going 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 ) X1F0 = 0, F1 = 1= XThusF0 1 + F1 X1 + F2 X2 + … + Fn-1 Xn-1 + Fn Xn + … X X 1 – X – X2== X/(1- X)(1 – (-)-1 X) -1 = 1, --1=1XX (1 – (1 – X)(1- (-X)(1- (-))--11X)X) Linear factors on the bottomn=0..∞=Xn?(1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn + …..) = 11 (1 – aX)(1-bX)(1 – aX)(1-bX) Geometric Series (Quadratic Form)aan+1 n+1 – – bbn+1n+1 aa - b- b n=0..∞Xn==11 (1 – (1 – X)(1- (-X)(1- (---11X))X)) Geometric Series (Quadratic Form)n+1 n+1 – (-– (--1-1))n+1n+1 √√5 5 n=0.. ∞=XnXX (1 – (1 – X)(1- (-X)(1- (--1-1X)X) Power Series Expansion of Fn+1 n+1 – (-– (--1-1))n+1n+1 √√55 n=0.. ∞=Xn+10 1 2 30 1 2 3201iiixF x F x F x F x F xx x L201 115ii iixxx xff�=� �� �� �= - -� �� �- -� �� ��The ith Fibonacci number is:01 15iiiff�=� �� �� - - �� �� �� �� ��Leonhard Euler (1765) J. P. M. Binet (1843)A de Moivre (1730)(1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn + …..) = 11 (1 – aX)(1-bX)(1 – aX)(1-bX) Let’s Derive Thisaan+1 n+1 – – bbn+1n+1 aa - b- b n=0..∞Xn==1 + Y1 + Y11 + Y + Y22 + Y + Y 33 + … + Y + … + Ynn + ….. = + ….. = 11 1 - Y1 - Y Substituting Y = aX …1 + aX1 + aX11 + a + a22XX22 + a + a33X X 33 + … + a + … + annXXnn + ….. =+ ….. = 11 1 - aX1 - aX Geometric Series (Linear Form)(1 + aX1 + a2X2 + … + anXn + …..)
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