Generalization of Fibonacci sequence

Transcription

1 Generalization of Fibonacci sequence Etienne Durand Julien Chartrand Maxime Bolduc February 18th 2013 Abstract After studying the fibonacci sequence, we found three interesting theorems. The first theorem is the relation between the number of seeds required to have a sequence and the recurrence relation. The second theorem is a generalization of the recurrence relation of Fibonacci, whatever the seeds are. Finally, the last theorem that we discovered is an interesting identity in a sequence of number coming from a certain recurrence relation. 1 Introduction Leonardo Pisano Bigollo, most commonly known as Fibonacci, is a famous Mathematician and is considered by many as the most talented mathematician of the Middle Ages. In the modern world, he is known widely for the spreading of the Hindu-Arabic numeral system in Europe, and for the Fibonacci numbers. 1

2 Fibonacci first explained the concept of Fibonacci numbers in his book of calculations in 1202, named Liber Abaci [1]. The concept of this sequence of numbers was explained by a growing population of idealized rabbits, assuming that: a newly born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at one month, so that at the end of its second month a female can produce another pair of rabbits. The concept assumes that rabbits never die and a mating pair always produces one new pair of one male and one female every month from the second month on.[2] 1. In the second month, the rabbits are mate but there is still only one pair. 2. In the third month, the pair of rabbits produces a new pair of rabbits, so there is now two pairs. 3. In the fourth month, the original pair produces a new pair of rabbits, so there is now three pairs. 4. In the fifth month, the original pair produces a new pair of rabbits, and the pair of rabbits that was born 2 months before also produces a new pair of rabbits, so there is now five pairs. The number of pair of rabbits in the month n will then be equal to the number of pair in the month (n-1) plus in the month (n-2). F n is the number of rabbit at the end of month n. The recurrence formula is then: 2

3 Fn = Fn 1 + Fn 2 F0 = 0 F1 = 1 The sequence of Fibonacci numbers creates a sequence that is known as Fibonacci sequence. Here is the sequence of the 15 first Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,... Fibonacci numbers now have many applications and it s possible to see fibonacci numbers in nature. For example, these numbers can be related to sunflowers [6]. Also, Fibonacci numbers are present in other mathematical principle, like in the Pascal Triangle. [5] Figure 1: Fibonacci number sequence in Pascal s triangle There are several ways to generalize the Fibonacci sequence, you can change the initial values (seeds), change the coefficients of the recurrence relation or change the number of terms (or order) of the recurrence relation. In this paper, generalizations of Fibonacci sequence will be investigated. Indeed, in this paper, sequences coming from different recurrence relations and seeds will be investigated. There are a lot of theorems and proprieties concerning the Fibonacci sequence, and not that many generalizations for other sequences, that s why our results could be relevant. 2 Basics of the Generalizations The main objective of our researches was to generalize the Fibonacci sequence. To be able to understand the generalizations of this sequence, there 3

4 are several concepts that need to be explained. Seeds The seeds in a sequence of numbers are very important. In fact, they dictate the pace of the sequence of numbers. These are defined as the two first numbers of the sequence. For example, in the Fibonacci sequence, the seeds are: F 0 = 0 and F 1 = 1 which gives the following sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,... When only one of the seeds is changed, all the sequence is affected. For example, Edouard Lucas did create a new sequence of numbers that used the same recurrence relation than Fibonacci, by changing the seeds to L 0 = 2 and L 1 = 1: Recurrence relation 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521,... The recurrence relation in a sequence is a determining factor for the appearance of the structure of the sequence. In fact, it dictates how to add, subtract, multiply or divide the numbers. In the Fibonacci sequence, the recurrence relation is F n = F n 1 + F n 2. This recurrence means that if two consecutive numbers are added, the sum will give the value of the number that follows those two numbers. It is possible to make other sequences of numbers by changing the recurrence relation and keeping the same seeds. For example, changing the lucas sequence of L n = L n 1 +L n 2 by L n = L n 1 L n 2 will give: which is way different of : 2, 1, 2, 2, 4, 8, 32, 256, 8192, , 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521,

5 Closed-form formula Some time after the discovery of the Fibonacci numbers, a functional expression has been found to generate the Fibonacci numbers. This closed-form formula has a direct relation with the Golden ratio ϕ. Indeed, when you divide F n by F n 1, it will give a result that tends to ϕ.[3] lim x F n F n 1 = ϕ Figure 2: Fibonacci numbers in the golden spiral It s the mathematician Jacques Philippe Marie Binet [4] that has found the functional expression to find the Fibonacci numbers. The Binet Formula: F n = ϕn (1 ϕ) n 5 This formula has been proved but the proof is actually very long and complicated, so the formula will be tried with a random Fibonacci number to see if it works: n = 8 F n = ϕn (1 ϕ) n 5 5

6 F 8 = ϕ8 (1 ϕ) 8 5 F 8 = 46, , F 8 = 21 3 Results Theorem on the number of seeds required We have discovered that to generate a sequence of numbers with a recurrence relation, there is a certain number of seeds required to start the sequence. Indeed, to generate the Fibonacci sequence of numbers, which has a recurrence relation of F n = F n 1 + F n 2, the number of seeds you need to start with is two. In the case of Fibonacci sequence, the seeds are S 0 = 0 and S 1 = 1. If you have a recurrence relation of G n = G n 1 +G n 3, you can t generate the sequence of number G with only two seeds. This is because, at the beginning of the sequence, you won t be able to generate the next number. Consequently, to add G n 1 and G n 3, you need at least three consecutive numbers. Indeed, we have found that to generate a sequence of numbers using a recurrence relation, the number of seeds required is equal to the biggest index of this recurrence relation. Theorem 1. In a recurrence relation G n = G n 1 + G n y, the number of seeds required will be y if y > 1. This discovery will be very useful for the computer program because it will tell us how many seeds we need to have to produce a sequence with a certain recurrence relation. 6

7 Recurrence Theorem This is one of our first discovery on the Fibonacci numbers. The sum of three consecutive numbers in the Fibonacci sequence, added by the sum of the three next consecutive numbers, gives the sum of the three next consecutive numbers. This expression can be written like this: (F n + F n+1 + F n+2 ) + (F n+1 + F n+2 + F n+3 ) = F n+2 + F n+3 + F n+4 From these additions you can make a new sequence of numbers. To start this sequence of numbers, you have to add the first three consecutive numbers of the Fibonacci sequence, which are F 1 = 1, F 2 = 1 and F 3 = 2. This will give the first number of the new sequence, that we will call G sequence. So G 1 = F 1 + F 2 + F 3. For the next numbers of the sequence, this simple formula can be applied G n = F n + F n+1 + F n+2. We can now write the sequence starting at G 1 : 4, 6, 10, 16, 26, 42, 68, 110, 178, 288,... From now on, you can easily see that this sequence of numbers uses the same recurrence relation than the Fibonacci sequence, which will be G n = G n 1 + G n 2. Also, if you continue the sequence further down, you can see that G 1 would be in reality G 3 and that the sequence has seeds of G 0 = 0 and G 1 = 2. This would be the complete sequence of numbers: 0, 2, 2,4, 6, 10, 16, 26, 42, 68, 110, 178, 288,... This discovery can be applied with the addition of four consecutive Fibonacci numbers, or five consecutive Fibonacci numbers, or even more of these numbers. In fact, this can be applied with whatever the quantity of consecutive numbers that you add of the Fibonacci sequence. In every case, these additions will give new sequences of numbers, that will all have the same recurrence relation of G n = G n 1 + G n 2, but having different seeds. The Recurrence theorem is the theorem that shows how all sequences that are made with addition of several consecutive numbers in Fibonacci sequence work. n is the quantity of consecutive Fibonacci numbers that you add to produce the sequence. c is the number of F that you start with. 7

8 Theorem 2. n+c 1 i=c F i + n+c i=c+1 F i = n+c+1 i=c+2 F i Indeed, the Recurrence theorem would be true n N and c N. Here is an example of the theorem by adding four consecutive Fibonacci numbers starting with F 2. n = 4 and c = i=2 F i + 5 F i + i=2 4+2 i=2+1 F i = 6 F i = i= i=2+2 (F 2 + F 3 + F 4 + F 5 ) + (F 3 + F 4 + F 5 + F 6 ) = F 4 + F 5 + F 6 + F = 29 The theorem can be proved by using the recurrence relation of the Fibonacci sequence F n = F n 1 + F n 2 : 7 i=4 F i F i Proof. n+c 1 i=c F i + n+c i=c+1 = (F c + F c F c+n 1 ) + (F c+1 + F c F c+n ) F i = F c + F c+1 + F c+n 1 + F c+1 + F c+2 + F c+n = F c+1 + 2F c+2 + F c+n 1 + F c+n = F c+2 + F c+3 + F c+n 1 + F c+n = F c+2 + F c F c+n+1 = n+c+1 i=c+2 F i 8

9 The Recurrence theorem can be even more generalized by applying it on other sequences than the Fibonacci sequence. In fact, we have found that the theorem works not only with the Fibonacci sequence of numbers, but with every sequence of numbers using the recurrence formula G n = G n 1 + G n 2, whatever the seeds are. Considering n and c are natural numbers, the Recurrence theorem is now: Theorem 3. n+c 1 i=c G i + n+c i=c+1 G i = n+c+1 i=c+2 G i Multiplication theorem We found an interesting relation in the Fibonacci sequence when the recurrence relation in change by: G n = G n 1 + 2G n 2 with the same seeds as Fibonacci: G 0 = 0, G 1 = 1. The sequence of numbers is given by: 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, By analyzing this sequence, an interesting recurrence in the numbers can be seen. In fact, by adding all the numbers as a summation, we get the following number of the sequence. However, adding the previous number will not always give the number that follows. When the n = even number, it is always necessary to add 1 to the sum. The theorem that will give the number is given by: Theorem 4. n 1 if n is odd then G i + 1 n 1 if n is even then By using the induction method, we can prove that it is true n N: G i Proof. G 1 = G = 1 G 1 = 1 9

10 2k 2 we assume that G 2k 1 = 1 + then, 1 + 2k G 1 = 1 + 2k 2 = G 2k 1 + G 2k + G 2k 1 = G 2k + 2G 2k 1 G i G i + G 2k + G 2k 1 = G 2k+1 (the next odd number) It is possible to prove the theorem for even numbers: Proof. then, n 1 G n = suppose, G 2k = 2k 1 G i = 2k 2 G i 2k 1 G i G i + G 2k+1 = G 2k+1 + G 2k = G 2k+2 (the next even number) BCD.html BCD is a program created using the html interface and coded mainly in JavaScript. This program eases the calculations of the F n = af n 1 +bf n c sequence with multiple selectable seeds and selectable coefficients of a, b and c. You can also change the number of iterations the program calculates. The core mechanics of the program is based on a for() loop that repeats this formula for its number of times, depending on user input: Where a, b and c correspond to the coefficients in the initial formula. First we create an array called fib[]. In arrays you can organize data by position, 10

11 so if we have an array called: example (1,5,3,7), example[0] would correspond to the first number in the array, 1, example[1] would be the second, 5, and so on. To include seeds in the calculation, numbers are added beforehand in the fib[] array. When the for() loop calculates the fib[] array, it will use the initial data provided in the array to create the sequence. A for() loop executes code that is written in between the braces, while using the parameters provided in the (). In this case, the parameters are (i=c; i its; i++). These are fairly simple and guide the for() loop to proceed the following way : execute the code contained into braces for its number of times (default is 20 iterations), while using c number of seeds, which corresponds to the Theorem on the number of seeds required. After executing the formula, the next step is to push the calculated data in the fib[] array into the second array, sequence[] which contains the entire sequence. It is required to do this, since the data in the fib[] array is only temporary, and each time it loops around, it overwrites the old data with the next number in the sequence. On each calculation of the loop, the data is pushed into the second array, where everything is kept, and then later displayed on the text area.?when developing BCD.html, we encountered many problems perfecting the equations, user input formulas, display texts and the core mechanics. The first issue was to integrate the source equation into JavaScript. We knew little on how to proceed, but we used a for() loop to calculate the equation, since it permitted me to control certain parameters and the number of iterations the program pushed into the text area. After the equation integration was done, the core essentials were there, 11

12 Figure 3: BCD program we had 2 seeds, F 0 and F 1, a recurring formula, and an array containing the output of said formula. The next issue was displaying the results in the array to the user of the program. To do so we initially used a input box to display the results. The input box displays the results on only one line. It was fine for 20 or so iterations, but when you increased these iterations, the numbers would go out of the input box. That was not convenient at all, so we searched for a better way to display the results. TextArea was the answer. TextArea permitted me to arrange the data in a more efficient way, since we could code-in the number of columns and rows that we wanted. So now, unless you have over 300 iterations, the numbers will not go out of the TextArea. Next challenge was integrating a modifiable core formula with two coefficients and interchangeable seeds that are selectable by the user. The coefficients were easy to integrate because they can be assigned to variables that are added into the core formula. The seeds, on the other hand, required a lot of fiddling around to get them to work. Initially we thought that only adding more initial values to the array would do the trick, but that was not 12

13 working at all, completely corrupting the data. We forgot to change a parameter in the for() loop, and when we found that out, everything worked in harmony. Using the Theorem on the number of seeds required we could link directly the number of seeds required to the c coefficient. Lastly, when we finished the program, we decided to rewrite some parts of it into a more efficient code, by using more arrays and more for() loops for different parts of the program. Due to time constraints, we could not finish coding the more efficient code by time. But this affects in no way the accuracy, speed or the validity of the calculated data. To finish, our program contains a couple of limitations. First is the number of seeds available for the user to change. Currently, you can only change the first 20 seeds of the core formula. We could of coded more inputs, but the program would have displayed in a very disorganized manner, and so we decided to keep it to ten. If there is any need for more than 20 seeds, it is an easy fix. When displaying iterations, there are three problems; first is the when you have more than a couple hundred iterations displaying on the TextArea, the data goes out of bounds. Secondly, when working with very large numbers, JavaScript will display them in a similar fashion: e Obviously this number is out of our understandable reach of comprehension, and optimally, we would like the program to stop the iterations when these numbers start appearing, but that is out of my programing skills range. Lastly, the data doesnt show up in a nice, elegant vertical form, but rather in a horizontal form, separated by commas. Optimally, the data should be displayed with a line break after each number, but it is very complicated to integrate text-based string elements into integer-filled arrays. 4 Conclusion There are three main discoveries that we made during our researches. First of all, we found that there was a certain number of seeds that were required to make a sequence using a certain recurrence relation. In fact, this number corresponds to the biggest index of the recurrence relation. In a recurrence relation G n = G n 1 + G n y, the numbers of seeds required will always be y. 13

14 Secondly, we have found a theorem that is true for all sequences of numbers that use the same recurrence relation then the Fibonacci sequence G n = G n 1 + G n 2, whatever the seeds are. This theorem says that the of a quantity of consecutive numbers n, starting with the c number in the sequence, plus the of the same quantity of consecutive numbers, starting with the c + 1 number in the sequence, will give the of the same quantity of consecutive numbers, starting with the c + 2 number in the sequence. Thirdly, we have found an interesting theorem. When the Fibonacci recurrence relation is change for F n = F n 1 + 2F n 2. The sequence is given by the sum of each previous numbers and adding 1 to this summation when n is odd. Few problems were encountered in our researches. First of all, some proofs were hard to find. Specially the proof for the Multiplication theorem because of the induction method. The second part that caused problems was all the creation of pictures and diagrams and even to find a suitable location for those pictures and diagrams. Concerning the Recurrence theorem, there is a clear path for future investigations. The main goal would be to find another theorem that will generalize every sequence of numbers, whatever the recurrence relation and the seeds are. There is also a path of future investigation with the Multiplication theorem. By changing the seeds of the sequence used and by seeing if the theorem still works, there is a possibility to generalize it. Another path of future investigation on generalization would be to see what happens if in the recurrence relation you change the addition for a multiplication, and to try it with different seeds. 14

15 5 Acknowledgements We would like to thanks Maxime Desbiens for helping us with the beginning of our project. Indeed, he helped for the conceptualization of the Recurrence theorem. Also, we are thankful to Patrick St-Amant, who believed in us, and helped us with the proof of several conjectures. Finally, thanks to Tamara Snyder Caron for correcting some english mistakes. References [1] Boyer, Carl B.. A history of mathematics. New York: Wiley, [2] Ouaknin, Marc. Mystres des chiffres. Nouv. d. ed. Paris: Assouline, [3] Bernois, Jean. Le nombre d or. Paris: Trajectoire, [4] Cohen, Gilles. Grands mathematiciens modernes. Paris: Ed. Pole, [5] Fibonacci Number. mathworld. N.p., n.d. Web. 13 Feb mathworld.wolfram.com/fibonaccinumber [6] The Golden Ratio and Fibonacci Numbers Math is Fun - Maths Resources. N.p., n.d. Web. 14 Mar