Pascal triangle variation and its properties

Transcription

1 Pascal triangle variation and its properties Mathieu Parizeau-Hamel 18 february 2013 Abstract The goal of this work was to explore the possibilities harnessed in the the possible variations of the famous Pascal triangle and in the incorporation of the Fibonacci and Lucas number sequences. Many relations of interest were found and the results helped to understand mathematical principles such as number sequence recurrence relations and seed numbers. Most of the relations present in the triangle remain to be found and this opens the door to number series never before seen. 1 Introduction In mathematics, the Pascal triangle was named after Blaise Pascal as he was the first to develop a significant quantity of uses for the triangular array of binomial coefficients [1, 2, 3]. Starting with an apex of 1 with positions outside the triangle counting as 0 and other numbers constituted of the sum of the above left and right numbers, one of the most well-known relations in the Pascal triangle is the Fibonacci series relation. When the numbers in the diagonals shown in Figure 1 were added, the results of the sums were to be the Fibonacci sequence [4, 5]. The Fibonacci sequence was named after Leonardo Pisano Bigollo, an Italian mathematician of the middle ages which is also responsible for the spreading of our present numeral system throughout Europe [8, 9]. It can be found in any subject that respects the golden ratio. One of these is the Fibonacci tiling which is obtained by adding a square to the right side of another square which side will be as big as the sum of all the sides of the 1

2 other squares it touches. The numbers that represent the length of the side of each square are found to be the fibonacci numbers. They are defined by the recurrence relation With seed values of The first numbers of this sequence are listed below F n = F n 1 + F n 2 (1) F 0 = 0, F 1 = 1 (2) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, (3) These numbers can be generalized to obtain a different sequence of numbers called the Lucas numbers which employ the same relation [6, 7] With seed values of And are found in the integer sequence below L n = L n 1 + L n 2 (4) L 0 = 0, L 1 = 1 (5) 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, (6) Following the dynamic of the Pascal triangle, the Fibonacci numbers and of the Lucas numbers, two new triangles were created with different addition arrangements, one with the Fibonacci sequence and another with the Lucas sequence. The goal was to see what sequences of numbers would come out of such arrangements, if any relations could be observed out of these and to better represent the Fibonacci relation and its generalizations such as the Lucas sequence in the conditions of the Pascal triangle variation. 2

3 2 The basics of the Pascal triangle and of its variation The principle of seed number and recurrence relation along with the understanding of the Pascal triangle variation rules and addition arrangement is critical to the good understanding of the results. These factors are closely related to one another and their individual or common modification can alter each other, resulting in interesting or unwanted results. 2.1 Pascal triangle variation rules and addition arrangement Unlike the original version of the Pascal triangle where one number was the sum of the above left and right numbers, the variation presents a different approach to the shape medium of the triangle. First of all the Fibonacci numbers are placed at the left of the triangle in increasing order so that the apex of the triangle is equal to F 0 as depicted in Figure 1 Then, the triangle is completed such as depicted in figure 1 and 2 for the variation of Fibonacci and Lucas respectively. This process is repeated throughout the shape. The number sequence can be changed with any generalization of the Fibonacci sequence as long as the recurrence relation (1) or (4) are respected for most of the results below to remain true. The Pascal triangle variation that is found reveals interesting properties that will be further explained in the result section of this paper 2.2 Fibonacci and Lucas seed numbers The Fibonacci series use the seed numbers F 0 = 0 and F 1 = 1 to initiate the whole sequence (3). If any of these numbers were to change, the sequence is likely to become very different. It is indeed what Edouard Anatole Lucas did by changing the seeds to F 0 = 2 and F 1 = 1 and gave what is now known as the Lucas sequence (6). For example, if the seed numbers of the recurrence relation G n = G n 1 +G n 2 3

4 Figure 1: Pascal triangle variation rules with the Fibonacci sequence 4

5 Figure 2: Pascal triangle variation rules with the Lucas sequence 5

6 were to be G 0 = 1 and G 1 = 1 the following integer sequence would be found And so, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, (7) For F n having the interger sequence depicted in (3). G n = F n+1 (8) 2.3 Fibonacci and Lucas recurrent relation Although the different seeds numbers of the Fibonacci and Lucas sequences produce some quite different number suites, their recurrent relation is the same. What it does is adding the two numbers to the left of the desired number starting with the sum of the given seeds. It is also possible to change the recurrence relation in order to provide a different number sequence that differs from the Fibonacci and Lucas sequences. For example, the recurrence relation H n = H n 1 H n 2 (9) For seed numbers seen in (2) would give the integer sequence 0, 1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, (10) 2.4 Bisection of a number sequence The bisection of a number sequence only contain one part of the original number sequence. The Fibonacci and Lucas number sequences are respectively defined such as F 2 n ; F 2 n 1 (11) and L 2 n ; L 2 n 1 (12) 6

7 3 Results While completing the Pascal triangle variation and exploring its properties, newfound number sequences and the their attributes, many interesting results were found and are presented below. 3.1 Pascal triangle variation properties One of the most apparent properties about this variation is present as long as the recurrent relations (1) or (4) are respected. In this sense, both Fibonacci an Lucas sequence will express the same properties in this subsection. When the addition arrangement is respected, the Fibonacci sequence or any generalization of its seeds repeats itself to give the number sequences depicted in Figure 3. Figure 3: Pascal triangle variation property with the Fibonacci sequence 7

8 The the sum of each number present in individual latitudes of the triangle can be expressed in the form below n F 2i 1 i=1 n F i 1 (13) i=1 Following the principles of this relation, it can be said that every number in the Pascal triangle variation is part of the number sequence that initiated the triangle since adding to consecutive numbers in these sequences will give the next number. The geometry of the triangle makes this possible even though it can be represented in altered forms. 3.2 Fibonacci and Lucas bisection number sequences Many number sequences can be found in the triangle but both bisections of the Fibonacci or Lucas sequences, depending on what sequence was used in the making of the triangle, can be found repetitively like depicted in Figure 4 which directly links the triangle variation to a new window of possibilities. For the Lucas sequence, the following bisections can be found 2, 3, 7, 18, 47, 123, 322, 843, 2207, (14) 1, 4, 11, 29, 76, 199, 521, 1364, 3571, (15) For the Fibonacci sequence, the following bisections can be found 0, 1, 3, 8, 21, 55, 144, 377, 987, (16) 1, 2, 5, 13, 34, 89, 233, 610, 1597, (17) 3.3 Pascal triangle variation sequence conjecture An important conjecture is that numbers repeat themselves in the triangle in straight diagonal lines such as depicted in Figure 5 and the order of these lines respects the one of the initial number sequence. The sum of each of these lines could be expressed in the following if n is an even number and the recurrent relation is the same as (1) and (4) ( n 2 + 1) F n; ( n 2 + 1) L n (18) 8

9 Figure 4: Pascal triangle variation with the Fibonacci bisection sequences 9

10 And if n is an odd number ( n ) F n; ( n ) L n (19) If the initial sequence used in the making of the triangle is the Fibonacci sequence, this number sequence will result And for the Lucas sequence 0, 1, 2, 4, 9, 15, 32, 52, 105, (20) 2, 1, 6, 8, 21, 33, 72, 116, 235, (21) These sequences may differ significantly from their original number sequences but are closely related to them. The Pascal triangle variation tremendously help to show such relations. 10

11 Figure 5: Pascal triangle variation relation with the Fibonacci sequence 11

12 4 Conclusion Starting from 3 classic concepts of discrete mathematics including the Pascal Triangle, the Fibonacci sequence and the Lucas sequence, a new approach was adopted to further investigate the variation of the well known Pascal triangle while directly including the Fibonacci series and one of it s most popular generalizations, the Lucas series. After the variation was completed, it showed relevant properties absent from the original triangle. The first one of these was the presence of the selfrepetition of the sequence used to complete the variation present in a partial form defined in the result section (11). This lead to the conclusion that this representation was of great value to better understand the workings of the Fibonacci sequence recurrent relation F n = F n 1 + F n 2. It also lead to the discovery that this relation was true for any seed of this same relation, meaning it was also true for the Lucas series. Afterwards, it also became apparent that the triangle harnessed sequences already covered by other papers such as the Fibonacci and Lucas Bisection number sequences which only included one half of their original Fibonacci and Lucas series. Also, newfound sequences were explored such as the 3.3 sequence given by straight lines featuring the same Fibonacci of Lucas numbers. The triangle helped to show how much the differing sequences obtained were related to the original sequence used in the making of the triangle. Before interesting number series were found, the search for relations within them proved to be very difficult. The triangle might not have a direct applications but it helps to show many mathematical concepts. Since only a few sequences and relations have been studied, it would be greatly interesting to investigate the several number series that have yet to be studied and the possible relations they may have. Also, the geometry of the numbers in the triangle remain of great interest and could have a potential for future researches. 12

13 5 Acknowledgements I would like to thank Julien Chartrand and Emile Boily-Auclair for their guidance, help and moral support throughout the realization of this document. References [1] Peter Fox (1998). Cambridge University Library: the great collections. Cambridge University Press. p. 13. ISBN [2] A. W. F. Edwards. Pascal s arithmetical triangle: the story of a mathematical idea. JHU Press, Pages [3] Gale, Thomson. Research Article: Pascal s Triangle. BookRags. BookRags, n.d. Web. 17 Feb [4] Goonatilake, Susantha (1998). Toward a Global Science. Indiana University Press. p ISBN [5] Singh, Parmanand (1985). The So-called Fibonacci numbers in ancient and medieval India. Historia Mathematica 12 (3): doi: / (85) [6] Goonatilake, Susantha (1998). Toward a Global Science. Indiana University Press. p ISBN [7] Chris Caldwell, The Prime Glossary: Lucas prime from The Prime Pages. [8] The Fibonacci Series - Biographies - Leonardo Fibonacci (ca ca.1240). Library.thinkquest.org. Retrieved [9] Leonardo Pisano - page 3: Contributions to number theory. Encyclopdia Britannica Online, Retrieved 18 September