fibonacci sequence

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CCC801, ART OF NUMBERS

NEELANSH KULSHRESHTHA

SNU ID: 1510110241

FIBONACCI SEQUENCE

WHAT IS “FIBONACCI SEQUENCE”:-

The Fibonacci numbers or Fibonacci sequence are the numbers in the

following integer sequence:-

0, 1, 1, 2, 3, 5, 8, 13, 21, 34,

!r

1, 1, 2, 3, 5, 8, 13, 21, 34,

"# $efinition, the first two numbers in the Fibonacci sequence are

either 1 an$ 1, or 0 an$ 1, $e%en$ing on the chosen starting %oint of

the sequence

The ne&t number is foun$ b# a$$ing u% the two numbers before it

• The 2 is foun$ b# a$$ing the two numbers before it '1(1)

• *imilarl#, the 3 is foun$ b# a$$ing the two numbers before it

'1(2),

• +n$ the 5 is '2(3), an$ so on

ere is a longer list:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 8., 144, 233, 3//, 10, .8/, 15./,2584, 4181, /5, 10.4, 1//11, 285/, 438, /5025, 1213.3,

1.418, 31/811,

T :-

n mathematical terms, the sequence Fn of Fibonacci numbers is

$efine$ b# the recurrence relation





SNU ID: 1510110241

First, the terms are numbere$ from 0 onwar$s li6e this:

n 7 0 1 2 3 4 5 / 8 . 10 11

Fn 7 0 1 1 2 3 5 8 13 21 34 55 8.

So term number 6 is called f6 (which equals 8).

There is an interesting %attern:

• oo6 at the number f 3 = 2 er# 3r$ number is a multi%le

of 2 '2, 8, 34, 144, 10, )

• oo6 at the number f 4 = 3 er# 4th number is a multi%le

of 3 '3, 21, 144, )

• oo6 at the number f 5 = 5 er# 5th number is a multi%le

of 5 '5, 55, 10, )

+n$ so on 'eer# nth number is a multi%le of Fn)

T9* "! ;!:-

The sequence below <ero has the same numbers as the sequence

aboe <ero, e&ce%t the# follow a (-(- %attern t can be written li6e

this:

F=n 7 '=1)n(1 Fn

hich sa#s that term >-n> is equal to '=1)n(1 times term >n>, an$ the

alue '=1)n(1 neatl# ma6es the correct 1,-1,1,-1, %attern

The sequence wor6s below <ero also, li6e this:

n 7 - -5 -4 -3 -2 -1 0 1 2 3

4





SNU ID: 1510110241

Fn 7 -8 5 -3 2 -1 1 0 1 1 2 3

?

REAL WORLD APPLICATION OF

“FIBONACCI SEQUENCE”

The Fibonacci *equence as man# real worl$ a%%lication an$ the coer

a wi$e range of %henomenon t@s foun$ in man# %laces li6e:-

F"!A+BB *CAB A A+T:-

The Fibonacci numbers are AatureDs numbering s#stem The# a%%ear

eer#where in Aature, from the leaf arrangement in %lants, to the

%attern of the florets of a flower, the bracts of a %inecone, or the scales

of a %inea%%le The Fibonacci numbers are therefore a%%licable to the

growth of eer# liing thing, inclu$ing a single cell, a grain of wheat,

a hie of bees, an$ een all of man6in$

The Fibonacci *equence@s a%%lications are not onl# limite$ to nature

but the# are also a%%lie$ in man# algorithmic techniques





SNU ID: 1510110241

The most common e&am%le of these algorithmic a%%lications are:-

The Fibonacci ea%s an$ The Fibonacci *earch Technique

!ut of these two, The Fibonacci *earch technique is quite fascinating

Th F!"#n$%%! S$&%h T%hn!'(

n com%uter science, the Fibonacci search technique is a metho$ of

searching a sorte$ arra# using a $ii$e an$ conquer algorithm that

narrows $own %ossible locations with the ai$ of Fibonacci numbers

Bom%are$ to binar# search, Fibonacci search e&amines locations

whose a$$resses hae lower $is%ersion Therefore, when the elements being searche$ hae non-uniform access memor# storage 'ie, the

time nee$e$ to access a storage location aries $e%en$ing on the

location %reiousl# accesse$), the Fibonacci search has an a$antage

oer binar# search in slightl# re$ucing the aerage time nee$e$ to

access a storage location

Fibonacci search was first $eise$ b# )$%* +!f& '1.53) as a

minima& search for the ma&imum 'minimum) of a unimo$al functionin an interal

+E!T9:-

et k be $efine$ as an element in F , the arra# of Fibonacci

numbers n 7 F m is the arra# si<e f the arra# si<e is not a Fibonacci

number, let F m be the smallest number in F that is greater than n

The arra# of Fibonacci numbers is $efine$ where F k (2 7 F k (1 ( F k ,

when k 0, F 1 7 1, an$ F 0 7 0

n *im%le or$s: -

e $ii$e the arra# into two subsets accor$ing to the both %rece$ing

Fibonacci numbers an$ com%are DitemD to the element at the %osition

Fn-2 f DitemD is greater than the element, we continue in the right

subset, otherwise in the left subset n this algorithm we $onDt hae to





SNU ID: 1510110241

ma6e a $iision to calculate the mi$$le element, but we get along

onl# with a$$itions an$ subtractions n our im%lementation we

assume$ that the Fibonacci numbers are gien e&%licitl# 'eg as

constants in the frame %rogram)

To test whether an item is in the list of or$ere$ numbers, follow these

ste%s:

1 *et k 7 m

2 f k 7 0, sto% There is no matchG the item is not in the arra#

3 Bom%are the item against element in F k =1

4 f the item matches, sto%

5 f the item is less than entr# F k =1, $iscar$ the elements from

%ositions F k =1 ( 1 to n *et k 7 k = 1 an$ return to ste% 2

f the item is greater than entr# F k =1, $iscar$ the elements from %ositions 1 to F k =1 enumber the remaining elements from 1

to F k =2, set k 7 k = 2, an$ return to ste% 2

The written %rogram in B can be li6e this:-

Hinclu$e Ist$iohJ

int ricercaKfib'int aLM, int n, long &)

N int inf70, %os, 6G

static int 667 -1, nn7-1, fibLM7N0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 8., 144, 233, 3//, 10, .8/,

15./, 2584, 4181, /5, 10.4, 1//11, 285/, 438, /5025, 1213.3, 1.418, 31/811, 51422.,

832040, 1342., 21/830., 35245/8, 5/0288/, .22/45, 14.30352, 2415/81/, 3.0881.,

3245.8, 102334155, 15580141OG

if'nn7n)

N 670G

while'fibL6MIn) 6((G

6676G





SNU ID: 1510110241

nn7nG

O

else

6766G

while'6J0)

N %os7inf(fibL--6MG

if''%osJ7n)PP'&IaL%osM))G

else if '&JaL%osM)

N

inf7%os(1G

6--G

O

else N

return %osG

O

O

return -1G

O

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+s it can be seen that the aboe a%%lication of Fibonacci series is

reall# interesting an$ it has man# other similar a%%lications

"ut we must 6now that it is not a uniersal law:-

* 9 Bo&eter, has the following im%ortant quote:

QIt should be frankly admitted that in some plants the

numbers do not belong to the sequence of f's

[Fibonacci numbers] but to the sequence of g's [Lucas

numbers] or even to the still more anomalous





SNU ID: 1510110241

sequences

3,1,,!,",... or !,#,$,",16,...

%hus we must face the fact that it is reall& not a uni'ersal law butonl& a fascinatinl& re'alent tendency.R

*till, Fibonacci *equence is one of the most ama<ing Sart of numbers@

6nown to human 6nowle$ge to$a#

BIBLIO,RAPH:-

• wwwmathworl$wolframcom

• Fibonacci in Aature wwwwilsoncoeugae$u

• wwwmathssurre#acu6

• wwwmathsisfuncom

• wwwautotuwienacat

• wwwco$ei$ocom

• ntro$uction to Eeometr# '1.1, ile#, %age 1/2)

b# *9Bo&eter

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http://www.mathworld.wolfram.com/

http://www.jwilson.coe.uga.edu/

http://www.maths.surrey.ac.uk/

http://www.mathsisfun.com/

http://www.auto.tuwien.ac.at/

http://www.codeido.com/

http://www.jwilson.coe.uga.edu/

http://www.maths.surrey.ac.uk/

http://www.mathsisfun.com/

http://www.auto.tuwien.ac.at/

http://www.codeido.com/

http://www.mathworld.wolfram.com/

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