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The Fibonacci Numbers And An Unexpected Calculation.

Leonardo Fibonacci In 1202, Fibonacci proposed an issue about the development of rabbit populaces.

Leonardo Fibonacci In 1202, Fibonacci proposed an issue about the development of rabbit populaces.

Leonardo Fibonacci In 1202, Fibonacci proposed an issue about the development of rabbit populaces.

The rabbit proliferation show A rabbit lives perpetually The populace begins as a solitary infant match Every month, each profitable combine sires another combine which will get to be gainful following 2 months old F n = # of rabbit sets toward the start of the n th month

The rabbit generation display A rabbit lives always The populace begins as a solitary infant match Every month, each beneficial match brings forth another match which will get to be profitable following 2 months old F n = # of rabbit sets toward the start of the n th month

The rabbit propagation demonstrate A rabbit lives everlastingly The populace begins as a solitary infant match Every month, each profitable combine sires another combine which will get to be beneficial following 2 months old F n = # of rabbit sets toward the start of the n th month

The rabbit multiplication show A rabbit lives perpetually The populace begins as a solitary infant combine Every month, each beneficial combine sires another combine which will get to be gainful following 2 months old F n = # of rabbit sets toward the start of the n th month

The rabbit proliferation demonstrate A rabbit lives everlastingly The populace begins as a solitary infant match Every month, each profitable combine sires another match which will get to be gainful following 2 months old F n = # of rabbit sets toward the start of the n th month

The rabbit proliferation show A rabbit lives always The populace begins as a solitary infant combine Every month, each beneficial match sires another combine which will get to be gainful following 2 months old F n = # of rabbit sets toward the start of the n th month

The rabbit multiplication show A rabbit lives always The populace begins as a solitary infant combine Every month, each beneficial match conceives another match which will get to be beneficial following 2 months old F n = # of rabbit sets toward the start of the n th month

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

Inductive Definition or Recurrence Relation for the Fibonacci 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)

Sneezwort (Achilleaptarmica) Each time the plant begins another shoot it takes two months before it is sufficiently solid to bolster fanning.

Counting Petals 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia) 8 petals: delphiniums 13 petals: ragwort, corn marigold, cineraria, some daisies 21 petals: aster, dark peered toward susan, chicory 34 petals: plantain, pyrethrum 55, 89 petals: michaelmas daisies, the asteraceae family.

Pineapple whorls Church and Turing were both keen on the quantity of whorls in every ring of the winding. The proportion of sequential ring lengths approaches the Golden Ratio.

Bernoulli Spiral When the development of the living being is relative to its size

Bernoulli Spiral When the development of the living being is corresponding to its size

Let's enjoy a reprieve from the Fibonacci Numbers so as to comment on polynomial division.

1 + X + X 2 1 – X 1 - (1 – X) X 2 X 3 X - (X – X 2 ) - (X 2 – X 3 ) 1 + X + X 2 + X 3 + X 4 + X 5 + X 6 + X 7 + … = How to gap polynomials? 1 ? 1 – X …

X n+1 - 1 + X 1 + X 2 + X 3 + … + X n-1 + X n = X - 1 The Geometric Series

X n+1 - 1 + X 1 + X 2 + X 3 + … + X n-1 + X n = X - 1 The breaking point as n goes to endlessness of = - 1 X n+1 - 1 X - 1 X - 1 = 1 - X

1 + X 1 + X 2 + X 3 + … + X n + … .. = 1 - X The Infinite Geometric Series

1 + X 1 + X 2 + X 3 + … + X n + … .. = 1 - X ( X - 1 ) ( 1 + X 1 + X 2 + X 3 + … + X n + … ) = X 1 + X 2 + X 3 + … + X n + X n+1 + … . - 1 - X 1 - X 2 - X 3 - … - X n-1 – X n - X n+1 - … = 1

1 + X 1 + X 2 + X 3 + … + X n + … .. = 1 - X 1 + X 1 – X 1 - (1 – X) X 2 X 3 X - (X – X 2 ) + X 2 + … - (X 2 – X 3 ) …

X + X 2 + 2X 3 + 3X 4 + 5X 5 + 8X 6 1 – X – X 2 X - (X – X 2 – X 3 ) X 2 + X 3 - (X 2 – X 3 – X 4 ) 2X 3 + X 4 - (2X 3 – 2X 4 – 2X 5 ) 3X 4 + 2X 5 - (3X 4 – 3X 5 – 3X 6 ) 5X 5 + 3X 6 8X 6 + 5X 7 - (5X 5 – 5X 6 – 5X 7 ) - (8X 6 – 8X 7 – 8X 8 ) Something more muddled X 1 – X – X 2

Hence X = F 0 1 + F 1 X 1 + F 2 X 2 +F 3 X 3 + F 4 X 4 + F 5 X 5 + F 6 X 6 + … 1 – X – X 2 = 0 1 + 1 X 1 + 1 X 2 + 2X 3 + 3X 4 + 5X 5 + 8X 6 + …

Going the Other Way (1 - X - X 2 ) ( F 0 1 + F 1 X 1 + F 2 X 2 + … + F n-2 X n-2 + F n-1 X n-1 + F n X n + … = ( F 0 1 + F 1 X 1 + F 2 X 2 + … + F n-2 X n-2 + F n-1 X n-1 + F n X n + … - F 0 X 1 - F 1 X 2 - … - F n-3 X n-2 - F n-2 X n-1 - F n-1 X n - … - F 0 X 2 - … - F n-4 X n-2 - F n-3 X n-1 - F n-2 X n - … = F 0 1 + ( F 1 – F 0 ) X 1 F 0 = 0, F 1 = 1 = X

Thus F 0 1 + F 1 X 1 + F 2 X 2 + … + F n-1 X n-1 + F n X n + … X = 1 – X – X 2

I was attempting to make tracks in an opposite direction from them!

Sequences That Sum To n Let f n+1 be the quantity of various arrangements of 1's and 2's that aggregate to n. Illustration: f 5 = 5 4 = 2 + 2 2 + 1 + 1 1 + 2 + 1 1 + 1 + 2 1 + 1 + 1 + 1

Sequences That Sum To n Let f n+1 be the quantity of various arrangements of 1's and 2's that aggregate to n. f 2 = 1 = 1 f 1 = 1 0 = the void total f 3 = 2 = 1 + 1 2

# of arrangements starting with a 2 # of groupings starting with a 1 Sequences That Sum To n Let f n+1 be the quantity of various successions of 1's and 2's that entirety to n. f n+1 = f n + f n-1

Fibonacci Numbers Again Let f n+1 be the quantity of various arrangements of 1's and 2's that aggregate to n. f n+1 = f n + f n-1 f 1 = 1 f 2 = 1

Visual Representation: Tiling Let f n+1 be the quantity of various approaches to tile a 1 X n strip with squares and dominoes.

Visual Representation: Tiling Let f n+1 be the quantity of various approaches to tile a 1 X n strip with squares and dominoes.

Visual Representation: Tiling 1 approach to tile a segment of length 0 1 approach to tile a portion of length 1: 2 approaches to tile a piece of length 2:

f n+1 = f n + f n-1 f n+1 is number of approaches to title length n. f n tilings that begin with a square. f n-1 tilings that begin with a domino.

Let's utilization this visual representation to demonstrate two or three Fibonacci personalities.

Fibonacci Identities The Fibonacci numbers have numerous bizarre properties. The numerous properties that can be expressed as conditions are called Fibonacci personalities. Ex: F m+n+1 = F m+1 F n+1 + F m F n

m n m-1 n-1 F m+n+1 = F m+1 F n+1 + F m F n

(F n ) 2 = F n-1 F n+1 + (- 1) n

n-1 (F n ) 2 = F n-1 F n+1 + (- 1) n F n tilings of a portion of length n-1

n-1 n-1 (F n ) 2 = F n-1 F n+1 + (- 1) n

(F n ) 2 = F n-1 F n+1 + (- 1) n (F n ) 2 tilings of two pieces of size n-1

(F n ) 2 = F n-1 F n+1 + (- 1) n Draw a vertical "blame line" at the furthest right position (<n) conceivable without cutting any dominoes

(F n ) 2 = F n-1 F n+1 + (- 1) n Swap the tails at the blame line to guide to a tiling of 2 n-1 's to a tiling of a n-2 and a n.

(F n ) 2 = F n-1 F n+1 + (- 1) n Swap the tails at the blame line to guide to a tiling of 2 n-1 's to a tiling of a n-2 and a n.

n even (F n ) 2 = F n-1 F n+1 + (- 1) n-1 n odd

F n is known as the n th Fibonacci number F 0 =0, F 1 =1, and F n =F n-1 +F n-2 for n 2 F n is characterized by a repeat connection. What is a shut frame equation for F n ?

Leonhard Euler (1765) Binet

1.6180339887498948482045… ..

Golden Ratio Divine Proportion = 1.6180339887498948482045… "Phi" is named after the Greek stone carver Phi dias

Ratio of stature of the individual to tallness of a man's navel

Definition of (Euclid) Ratio got when you isolate a line section into two unequal parts with the end goal that the proportion of the entire to the bigger part is the same as the proportion of the bigger to the littler.

Expanding Recursively

Divina Proportione Luca Pacioli (1509) Pacioli dedicated a whole book to the sublime properties of . The book was outlined by a companion of his named: Leonardo Da Vinci

Table of substance The primary significant impact The second crucial impact The third solitary impact The fourth unspeakable impact The fifth honorable impact The 6th indescribable impact The seventh boundless impact The ninth most phenomenal impact The twelfth unique impact The thirteenth most recognized impact

Table of substance "For our salvation this rundown of impacts must end." .:tslidesep

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