# Ppt on fibonacci sequence

##### Fibonacci Born in: Pisa in 1175 Nickname: Fibonacci Real name: Leonardo Pisano or Leonardo of Pisa.

mathematics. Accomplishments u To become a famous mathematician. u He is widely known. u He introduced Arabic numerals to Europe. u He made the Fibonacci sequence. Bibliography Books: u Fascinating Fibonacci. Dale Seymour Publishing; Page 1 u Portraits;Dale Seymour Publishing: Fibonacci page CD Rom: u “Fibonacci” Encarta. Microsoft 1997. Internet-World Wide Web: u www.encarta.msn.com/index/concise/OVOL39/06D7 6000.asp

##### Lab 4 Design a processor to implement the Fibonacci sequence

will be given an instruction set You will design a datapath to implement the instructions You will design a FSM to implement an instruction sequence for the fibonacci sequence calculation Fibonacci Sequence Fibonacci sequence Pseudo code F(1) = 1 F(2) = 1 F(N) = F(N-1) + F(N-2) e/A + 0, carry_in = 1 Decrement can be implemented as A + “1111”, carry_in = 0 zero_flag = 1 if the output of the ALU is “0000” Fibonacci Sequence Fibonacci sequence Pseudo code F(1) = 1 F(2) = 1 F(N) = F(N-1) + F(N-2) e.g. 1, 1, 2,3,/

##### CMSC 150 RECURSION CS 150: Mon 26 Mar 2012. Motivation : Bioinformatics Example  A G A C T A G T T A C  C G A G A C G T  Want to compare sequences.

2) x … x 2 x 1  n! = n x (n-1)!  Defined recursively: 1if n = 0 n! =n(n-1)!if n > 0 Compute n! in BlueJ… Another Example : FibonacciFibonacci sequence:  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …  After first two, each term is sum of previous two  Defined recursively:  Let f n be the n th/ term, n = 0, 1, 2… 0if n = 0 f n =1if n = 1 f n-1 + f n-2 if n > 1 Another Example : FibonacciFibonacci sequence:  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, … 0if n = 0 f n =1if n = 1 f n-1 + f n-2 if n > 1  f/

##### The Golden Ratio and Fibonacci Numbers in Nature By: Mary Catherine Clark.

book Liber Abcai asked a question involving the reproduction of a single pair of rabbits which is the basis of the Fibonacci sequence.  It was posed: Suppose a newly born pair of rabbits (a male and female) are put in a /by!  As n increases, the ratio of approaches the golden ratio and is expressed as =  This is the fundamental property of both the Fibonacci sequence and the golden ratio.  Both of these ratios converge at the same limit and are the positive root of the quadratic equation  If the two/

The next example shows how to use the distinct-roots theorem to find an explicit formula for the Fibonacci sequence. 23 Example 4 – A Formula for the Fibonacci Sequence The Fibonacci sequence F 0, F 1, F 2,... satisfies the recurrence relation for all integers k  2 with/for all integers n  0. Remarkably, even though the formula for F n involves all of the values of the Fibonacci sequence are integers. cont’d 29 The Single-Root Case 30 The Single-Root Case Consider again the recurrence relation where A and/

##### Geometry: Similar Triangles. MA.912.D.11.5 Explore and use other sequences found in nature such as the Fibonacci sequence and the golden ratio. Block.

are 1 and 1, and each remaining number is the sum of the previous two. Fibonacci sequence: recurrence In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation with seed values Graph of consecutive values of Fibonacci sequence Golden mean Ratios of consecutive values of Fibonacci sequence: F n+1 /F n Tend to a number – it is Golden Mean Ratio of terms/

##### Fibonacci Numbers, Vector Programs and a new kind of science.

= f n + f n-1 Let f n+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 1 Leonardo Fibonacci In 1202, Fibonacci proposed a problem about the growth of rabbit populations. Rules 1.in the first month /(n) = Fib(n-1) + Fib(n-2) n01234567 Fib(n)0112358 1313 Fibonacci Numbers Again f n+1 = f n + f n-1 f 1 = 1 f 2 = 1 Let f n+1 be the number of different sequences of 1’s and 2’s that sum to n. Visual Representation: Tiling Let f/

##### CMPT 225 Recursion. Objectives Understand how the Fibonacci series is generated Recursive Algorithms  Write simple recursive algorithms  Analyze simple.

(21)  If we did then the answer is just fib(22) + fib(21)  What happens if we write a function to calculate Fibonacci numbers in this way? Calculating the Fibonacci Sequence Here is a function to return nth number in the Fibonacci sequence  e.g. fib(1) = 1, fib(3) = 2, fib(5) = 5, fib(8) = 21, … public static int fib(int n){ if(n/

##### THE FIBONOCCI SEQUENCE IN REAL LIFE BY ANNE-MARIE PIETERSMA, HARRY BUI, QUINN CASHELL, AND KWANGGEUN HAN.

the number of petals on a flower, generally you will find that this number is a Fibonacci Number, 3, 5, 8, 13, etc. However, this is just the beginning. The Fibonacci Sequence can also be seen when observing leaves. NATURE If you start at the bottom leaf / come to the next leaf that is in line with the initial starting leaf, generally the number will be in line with the Fibonacci Sequence. IN CONCLUSION: MATH AND SCIENCE ARE ALL AROUND US!! It’s easy to think (especially when students take classes for Gen Ed/

##### 6.Advanced Counting Techniques 1 Copyright M.R.K. Krishna Rao 2003 Ch 6. Recurrence Relations A recurrence relation for the sequence {a n } is an equation.

Techniques 8 Copyright M.R.K. Krishna Rao 2003 Rabbit Reproduction MonthAdult pairsYoungTotal 1011 2011 3112 4123 5235 6358 Fibonacci defined a sequence when observing the reproduction of rabbits. Starting with one male/female pair of newborn rabbits, he came up with/ pairs of mature rabbits will there be? 6.Advanced Counting Techniques 9 Copyright M.R.K. Krishna Rao 2003 Rabbit Reproduction - Fibonacci numbers This sequence of numbers {f n } satisfies the recurrence relation f n = f n-1 + f n-2, n  3 /

##### Spiral Growth in Nature

to the sum of the previous two. Each Fibonacci number has its place in the Fibonacci sequence. The standard mathematical notation to describe a Fibonacci number is an F followed by a subscript indicating its place in the sequence. For example, F8 stands for the eighth Fibonacci number, which is 21 (F8 = 21). Fibonacci Numbers Fibonacci numbers that come before FN, are FN-1 and FN-2. The/

#####  The Fibonacci Spiral is created from the Fibonacci Sequence. The Fibonacci sequence is a mathematical succession of numbers in which each number is.

many different forms such as flower patterns and shells. One of the best representations is in the sunflower. The pattern of seeds within a sunflower follows the Fibonacci sequence.  Another well known representation of the Fibonacci sequence is found in pinecones. In In the picture, there are 8 spirals bending to the left as they open out indicated in red. Likewise there are 13/

##### DUYGU KANDEMİR 200822024. CONTENT Who was Fibonnaci? Fibonacci Sequence Fibonacci Rectangle and Spiral Golden Ratio Fibonacci Rabbit Problem Pascal Triangle.

mathematician who lived 800 years ago!!! These numbers are to be found everywhere in nature. Applications of Fibonacci series are nearly limitless. Lots of matematicians added a new piece to the Fibonnaci puzzle. Fibonacci mathematics is a constantly expanding branch of number theory. 2) Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21, ………….. The first two numbers in the series are one and/

#####  2002 Prentice Hall. All rights reserved. 1 Week 2: Methods Questions about last week’s revision –C#.NET –.NET Framework –sequence, selection, repetition.

ends when the value is less than or equal to 1  2002 Prentice Hall. All rights reserved. 49 6.15 Example Using Recursion: The Fibonacci Sequence Fibonacci Sequence –F(0) = 0 –F(1) = 1 –F(n) = F(n - 1) + F(n - 2) –Recursion is / reserved. Outline 52 FibonacciTest.cs Program Output  2002 Prentice Hall. All rights reserved. 53 6.15 Example Using Recursion: The Fibonacci Sequence Fig. 6.17Set of recursive calls to method Fibonacci (abbreviated as F ). return 1return 0 F( 1 )F( 0 )return 1 F( 3 ) F( 2 )F(/

##### The Fibonacci Sequence. Leonardo Fibonacci (1170 – 1250) First from the West, but lots of evidence from before his time.

, 3, 5, 8, 13, … The current term is found by adding the previous two The Fibonacci Sequence Originated from a question he had about rabbit population The Fibonacci Sequence What’s special about it? If we continue the sequence… and take the limit of the ratio to infinity… The Fibonacci Sequence We find a ratio that is approximately 1.61803399 This is known as the GOLDEN Ratio/

##### Home Viruses Elizabeth Blackwell Why Learn French Good Fats, Bad Fats, and Worst Fats Basic Stroke Information Leonardo Pisano Fibonacci King Tut Helping.

decimal number system which replaced the Roman Numeral system.  It was this problem that led Fibonacci to the introduction of the Fibonacci Numbers and the Fibonacci Sequence which is what he remains famous for today. The sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.. This sequence is that each number is the sum of the two proceeding numbers. Home Viruses Elizabeth Blackwell/

##### 1 REVIEW OF ANALYSIS TECHNIQUES (UNIT-1). 2 1. ALGORITHM : is a sequence of unambiguous instructions for solving a problem in a finite amount of time.

(x,y), (x,y-x),… 25 The relationship between FN and GR : In the above diagram, we can see that y/x = x / (y-x)…………..(1) In Fibonacci Sequence : For large values of n, we have Fn / Fn-1 = Fn-1 / Fn-2 …..(2) 26 But, Fn = Fn-1 + Fn-2  Fn-2/ - Fn-1) If we replace, Fn with y and Fn-1 with x, we have y / x = x / (y – x) which is Golden Ratio. Compute any number in Fibonacci Sequence : Fn =  n / √5 where F0 = 0, F1 = 1, F2 = 1, F3 = 2,….. ***** 27 3. Recurrences & Solutions of Recurrence Equations : 3.1 Divide-/

##### Biography ( ) Fibonacci is a short for the Latin "filius Bonacci" which means "the son of Bonacci" but his full name was Leonardo of Pisa, or Leonardo.

spirals of the sunflower head pineapple scales Flowers Nautilus Sun Flower Pineapple in the introduction to Europe Conclusion The greatest European mathematician of the middle age, most famous for the Fibonacci sequence, in which each number is the sum of the previous two and for his role in the introduction to Europe of the modern Arabic decimal system. HAPPY EASTER!!! THE END/

##### Princeton University COS 423 Theory of Algorithms Spring 2002 Kevin Wayne Fibonacci Heaps These lecture slides are adapted from CLRS, Chapter 20.

node of size s k, and let y 1,..., y k be children in order that they were linked to x*. Assume k  2 41 Fibonacci Facts Definition. The Fibonacci sequence is: n 1, 2, 3, 5, 8, 13, 21,... Slightly nonstandard definition. Fact F1. F k   k, where  =/ Fact F2. Consequence. s k  F k   k. n This implies that size(x)   degree(x) for all nodes x. 42 Golden Ratio Definition. The Fibonacci sequence is: 1, 2, 3, 5, 8, 13, 21,... Definition. The golden ratio  = (1 +  5) / 2 = 1.618… n Divide a rectangle into/

##### Fibonacci Problem Solving and Thinking in Engineering Programming H. James de St. Germain.

the first ‘X’ Fibonacci Numbers Really Understand the Problem Start with 0 and 1 (by definition) Start of sequence is: 0,1 Add these two together: 1 Expanded sequence is:0,1,1 Add last two numbers together –1+1 = 2 Expanded sequence is: 0,1/ minutes to draw a black box for this function Function as Black Box Function InputOutput Fibonacci as Black Box Fibonacci Count Fibonacci Numbers Compute the first “count” fibonacci numbers Array of Numbers Integer Comment Your Function You have 1 minute to write a brief/

##### The Golden Ratio, Fibonacci Numbers, And Other Recurrences Great Theoretical Ideas In Computer Science John LaffertyCS 15-251 Fall 2006 Lecture 13October.

Sum To n f n+1 = f n + f n-1 Let f n+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 1 Fibonacci Numbers Again f n+1 = f n + f n-1 f 1 = 1 f 2 = 1 Let f n+/1 be the number of different sequences of 1’s and 2’s that sum to n. Visual Representation: Tiling Let f /

##### Expressing Sequences Explicitly By: Matt Connor Fall 2013.

they do not have a common constant or ratio These are commonly called Fibonacci-type The difficult thing about these is finding an explicit formula Now we will go through deriving an explicit formula for the Fibonacci Sequence We know the relational formula is A n = A n−1 + /the form A n =Cx n and plug it in to the relational equation and get Cx n = Cx n−1 + Cx n−2 Fibonacci Sequence Explicit Formula Cx n = Cx n−1 + Cx n−2 this will always simplify to an equation with the same coefficients as the relational /

##### SECTION 5-5 The Fibonacci Sequence and the Golden Ratio Slide 5-5-1.

how many pairs of rabbits will there be at the end of one year? THE FIBONACCI SEQUENCE Slide 5-5-3 The solution of the problem leads to the Fibonacci sequence. Here are the first thirteen terms of the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, /is obtained by adding the two previous terms. RECURSIVE FORMULA FOR FIBONACCI SEQUENCE Slide 5-5-4 If F n represents the Fibonacci number in the nth position in the sequence, then EXAMPLE: A PATTERN OF THE FIBONACCI NUMBERS Slide 5-5-5 Find the sum of the squares of/

##### A.) { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.. } These are some examples of sequences: B.) { 6, 3, 3, 2, 4, 1 } C.) { 3, 6, 12, 24, 48, 96, 192, 384, 768..

to multiply by two. B.) { 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.. } This is another example, called the Fibonacci sequence. Each element is the sum of the two before it; 5 = 3+2, 21= 13+8. In this case there are two values that this does not work for/just as many roots as initial values. So it makes sense that we work out the coefficients by using the initial values. Let’s find the coefficients for the Fibonacci sequence. We already know the roots, so: a n = b 1 [ (1+√5)/2] n + b 2 [(1- √5)/2] n If we plug/

##### Examining Fibonacci sequence right end behaviour 1,1,2,3,5,8,..............................., What is the limiting value for the ratio of consecutive terms.

,1,2,3,5,8,..............................., What is the limiting value for the ratio of consecutive terms in the Fibonacci sequence? To the far right, does this sequence behave like a geometric sequence? x1 x2 X1.5 For the left end behaviour, the first few numbers, the Fibonacci Sequence does not appear to be of a geometric type, but perhaps the right end behaviour, the list does/

##### The Art of Fibonacci Trading

and facts when he published his findings When we think of Fibonacci, we think of his introduction on the Hindu-Arabic numerals (HAN) to the Western world and the famous Fibonacci sequence MORE ABOUT FIBONACCI Fibonacci is one of general revelation from God to mankind that gave /.618) didapat dengan membagi angka setelah deret ke 13 dengan angka sebelumnya 233/144 = 1,618 377/233 = 1,618 610/377 = 1,618 Fibonacci Sequence and Ratio 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584/

##### Fibonacci Heaps CS 252: Algorithms Geetika Tewari 252a-al Smith College December, 2000.

over all the operations performed. - Can be used to show that average cost of an operation is small, if you average over a sequence of operations, even though a single operation might be expensive Fibonacci Heap Operations 1. Make Fibonacci Heap -Assign N[H] = 0, min[H] = nil Amortized cost is O(1) 2. Find Min -Access Min[H] in O(1) time/

##### Basic Practice of Statistics - 3rd Edition

Basic Practice of Statistics - 3rd Edition Lecture 4 1. Fibonacci Sequence 2. Golden ratio Chapter 2 Basic Practice of Statistics - 3rd Edition Fibonacci Sequence Recap: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … Finding Patterns in Nature that exhibit numbers in the sequence: your project. Chapter 2 Basic Practice of Statistics - 3rd Edition Plants White Calla Lily One Petal Euphorbia Two/

##### CHAPTER 8 Dynamic Programming. Algorithm 8.1.1 Computing the Fibonacci Numbers, Version 1 This dynamic-programming algorithm computes the Fibonacci number.

1; f[n] = f[n - 1] + f[n - 2], n ≥ 3. At the conclusion of the algorithm, the array f holds the first n Fibonacci numbers. Input Parameters: n Output Parameters: None fibonacci1(n) { // f is a local array f[1] = 1 f[2] = 1 for i = 3 /1,j,denom,used) } Algorithm 8.3.1 Optimal Matrix Multiplication This algorithm computes the minimum number of scalar multiplications to multiply a sequence of n matrices. The input is the array size that contains the sizes of the matrices to be multiplied. The first matrix is /

##### The Sequence of Fibonacci Numbers and How They Relate to Nature November 30, 2004 Allison Trask.

and How They Relate to Nature November 30, 2004 Allison Trask Outline History of Leonardo Pisano Fibonacci What are the Fibonacci numbers?  Explaining the sequence  Recursive Definition Theorems and Properties The Golden Ratio Binet’s Formula Fibonacci numbers and Nature Leonardo Pisano Fibonacci Born in 1170 in the city-state of Pisa Books: Liber Abaci, Practica Geometriae, Flos, and Liber Quadratorum Frederick II’s challenge/

##### Aim: Arithmetic Sequence Course: Alg. 2 & Trig. Do Now: Aim: What is an arithmetic sequence and series? Find the next three numbers in the sequence 1,

If the pattern is extended, what are the next two terms? How is this sequence different from the famous Fibonacci sequence? 4, 8, 12, 16, 20, 24,... positive integers terms of sequence Aim: Arithmetic Sequence Course: Alg. 2 & Trig. Model Problem Write the rule that can be used in forming a sequence 1, 4, 9, 16,..., then use the rule to find the next three terms/

##### The Fibonacci Sequence Ava Boussy. What is the Fibonacci Sequence? The fibonacci sequence is a pattern of numbers that uses adding, so once you get it,

The Fibonacci Sequence Ava Boussy What is the Fibonacci Sequence? The fibonacci sequence is a pattern of numbers that uses adding, so once you get it, it’s easy. 1,1,2,3,5,8,13….. Doodle in Math http://www.youtube.com/ guy) Look Familiar? Pascals Triangle Of Course.It was in a different doodling in math class video. Pascal’s Triangle in made of the Fibonacci Sequence. Do It – Use words to make it Yearn-1 syllable Pine-1 Desire-2 how I feel-3 About the nintendo-5 Sitting in the Gamestop window-8 Homework See /

##### Fibonacci The Fibonacci Sequence The Golden Ratio.

lived during the 12 th century Spread the use of the hindu- arabic numbers in Europe Used the Fibonacci sequence in his book Liber Abaci And the Sequence? The first two terms of the sequence are 1 and 1 Each succeeding term is the sum of the previous two terms, or (for/, 3, 5, 8, 13, 21, 34 Applications Choose any two numbers between 1 and 10. Add these numbers like you would a Fibonacci sequence. The sum is… The Golden Ratio Suppose a and b are two consecutive terms of the FS. Then the next term would be a+/

##### Rabbits, Fibonacci and Market Rabbits and Market Behavior Fibonacci Number & Golden Ratio Retracement & Projection.

rabbits population grows again? How you calculate the growth and decay pattern? Leonardo Fibonacci Leonardo Fibonacci, a famous mathematician, born in Pisa, Italy around 1170, figured out the rabbit puzzle. Fibonacci sequence, or Number Add the last two numbers to get the next. 0, 1/ to these patterns? What has this got to do with investment on market? The Golden Ratio Divide any number in the Fibonacci sequence by the one before it, for example 55/34, or 21/13, and the answer is always close to 1.61803/

##### 10-Jun-15 Fibonacci Numbers A simple example of program design.

and while statements) into a working program The example we use is writing a program to compute and display a Fibonacci sequence 3 Fibonacci sequences A Fibonacci sequence is an infinite list of integers The first two numbers are given Usually (but not necessarily) these are 1 and/ numbers: 1 1 2 3 5 8 13 21 34 55 89 144... Let’s write a program to compute these 4 Starting the Fibonacci sequence We need to supply the first two integers int first = 1; int second = 1; We need to print these out: System.out/

##### 1 CSC 222: Object-Oriented Programming Spring 2012 recursion & sorting  recursive algorithms  base case, recursive case  silly examples: fibonacci,

huge redundancy we will look at better examples later, but first analyze these simple ones /** * Computes Nth Fibonacci number. * @param N sequence index * @returns Nth Fibonacci number */ public int fibonacci(int N) { if (N <= 2) { return 1; } else { return fibonacci(N-1) + fibonacci(N-2); } /** * Computes Greatest Common Denominator. * @param a a positive integer * @param b positive integer (a >= b) * @returns GCD of a and b/

##### Home Viruses Elizabeth Blackwell Why Learn French Good Fats, Bad Fats, and Worst Fats Basic Stroke Information Leonardo Pisano Fibonacci King Tut Helping.

decimal number system which replaced the Roman Numeral system.  It was this problem that led Fibonacci to the introduction of the Fibonacci Numbers and the Fibonacci Sequence which is what he remains famous for today. The sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.. This sequence is that each number is the sum of the two proceeding numbers. Home Viruses Elizabeth Blackwell/

##### Fermat’s Little Theorem Fibonacci Numbers Shirley Moore CS4390/5390 Fall 2013 September 10, 2013 1.

modular-arithmetic.html mod mods powermod OK to use powermod for Problem 5 on Homework 1 See also www.wolframalpha.comwww.wolframalpha.com 6 Fibonacci Sequence A man puts a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be/ many digits are in F 1000 and what are the first few digits? More Exercises 7.Investigate the ratio F n+1 /F n of consecutive Fibonacci numbers and try to identify the limit. 8.Investigate the sum F 0 + F 1 + … + F n. Find a formula for this /

##### Who was Fibonacci ? Greatest European mathematician of the middle ages Born in Pisa, Italy, the city with the famous Leaning Tower,~ 1175 AD Major contributions.

… Count the number of spirals. Spirals in a pine cone: clockwise and anti-clockwise And more Pascal’s Triangle Finding the n-th term of a Fibonacci Sequence Summation Formula For the first “n” numbers in the Fibonacci Sequence Fibonacci Sequence As the terms increase, the ratio between successive terms approaches 1.618 This is called the “golden ratio” –Ratio of human leg length to arm/

##### Finance and the Fibonacci Sequence By Benjamin R. Hull.

to calculate present value, compound interest, geometric series… – Rabbit problem First western appearance of Fibonacci sequence Liber Abaci Rabbits! The Golden Ratio The Fibonacci Sequence Technical Trading Pioneered by Charles Dow A Random Walk Down Wall Street – Burton G. /, 1994. Pictures and Other Sources: http://upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Fibonacci.jpg/220px-Fibonacci.jpg http://en.wikipedia.org/wiki/File:The_Parthenon_in_Athens.jpg http://www.mathacademy.com/pr/prime/articles/

##### Phi, Fibonacci, and 666 Jay Dolan. Derivation of Phi “A is to B as B is to C, where A is 161.8% of B and B is 161.8% of C, and B is 61.8% of A and C is.

Parthenon, Athens United Nations Building, New York The Fibonacci Sequence Leonardo Pisano Fibonacci discovered it in about 1202 A.D. He lived from 1170-1250 He first used the sequence to calculate the growth of a Rabbit population. The sequence is: 0, 1, 1, 2, 3, / is found by adding up the two previous numbers. The Fibonacci sequence is found in nature and is still used to predict growth patterns. The sequence in Nature: Flower Petals 123 5 813 2134 Fibonacci in the Real World “A group of rabbits mate at /

##### 1 Binomial heaps, Fibonacci heaps, and applications.

#(ones) Amortized(increment) = actual(increment) +  Amortized(increment) = 1+ #(1 => 0) + 1 - #(1 => 0) = O(1) ==> Sequence of n increments takes O(n) time 15 Binomial heaps - amortized ana.  (collection of heaps) = #(trees) Amortized cost of insert O(1) Amortized cost/ cuts)  (decrease-key) = O(1) - #(cascading cuts) ==> amortized(decrease-key) = O(1) ! 32 Fibonacci heaps (analysis) Cascading cuts and successive linking will pay for themselves. The only question is what is the maximum degree of a/

##### In this chapter you have been writing equations for arithmetic sequences so that you could find the value of any term in the sequence, such as the 100.

b) Then Avery wrote t(n) = 6n + 8. Help Collin write a recursive equation. 5-75. The Fibonacci sequence is a famous sequence that appears many times in mathematics. It can describe patterns found in nature, such as the number of petals on / mathematicians in India. The equation that describes the Fibonacci sequence can be written as: t(1) = 1 t(2) = 1 t(n + 1) = t(n) + t(n – 1) a)Write the first 10 terms of the Fibonacci sequence. b)Is the Fibonacci sequence arithmetic, geometric, or neither? c)Describe what/

##### MATHLETES 10-24-12. Fibonacci Numbers and “The Golden Ratio”

15, 21, 27, 33... Add 6 to the previous number S n = S n-1 + 6 Examples of sequences 1, 3, 9, 27, 81, 243... Multiply the previous number by 3 S n = 3*S n-1 The Fibonacci Sequence Start with the numbers 1 and 1, and apply the following rule: S n = S n-1 + S /n-2 In other words, the next term is found from adding up the previous 2 numbers The First Few Fibonacci Numbers 1, 1, 2, 3, 5, 8, 13/

##### © Boardworks Ltd 2005 1 of 68 A7 Sequences KS4 Mathematics.

+11+22+33+55+88+1313+2121+34 © Boardworks Ltd 2005 66 of 68 The n th term of the Fibonacci sequence The Fibonacci sequence is an example of a sequence for which the general term cannot be written in terms of its position in the sequence. To find any given term we have to know the value of the previous two terms. We can write/

##### 1 Binomial heaps, Fibonacci heaps, and applications.

00011 00100 00101 10 Amortized analysis (Cont.) On the worst case increment takes O(k). k = #digits What is the complexity of a sequence of increments (on the worst case) ? Define a potential of the counter: Amortized(increment) = actual(increment) +   (c) =/cascading cuts)  (decrease-key) = O(1) - #(cascading cuts) ==> amortized(decrease-key) = O(1) ! 49 Fibonacci heaps (analysis) Cascading cuts and successive linking will pay for themselves. The only question is what is the maximum degree of a node/