Pascal triangle variation and its properties

Mathieu Parizeau-Hamel

18 february 2013

Abstract

The goal of this work was to explore the possibilities harnessed inthe the possible variations of the famous Pascal triangle and in theincorporation of the Fibonacci and Lucas number sequences. Manyrelations of interest were found and the results helped to understandmathematical principles such as number sequence recurrence relationsand seed numbers. Most of the relations present in the triangle remainto 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 wasthe first to develop a significant quantity of uses for the triangular array ofbinomial coefficients [1, 2, 3]. Starting with an apex of 1 with positions out-side the triangle counting as 0 and other numbers constituted of the sum ofthe above left and right numbers, one of the most well-known relations inthe Pascal triangle is the Fibonacci series relation. When the numbers in thediagonals shown in Figure 1 were added, the results of the sums were to bethe Fibonacci sequence [4, 5].

The Fibonacci sequence was named after Leonardo Pisano Bigollo, an Italianmathematician of the middle ages which is also responsible for the spreadingof our present numeral system throughout Europe [8, 9].

It can be found in any subject that respects the golden ratio. One of theseis the Fibonacci tiling which is obtained by adding a square to the right sideof another square which side will be as big as the sum of all the sides of the

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other squares it touches. The numbers that represent the length of the sideof each square are found to be the fibonacci numbers.

They are defined by the recurrence relation

Fn = Fn−1 + Fn−2 (1)

With seed values ofF0 = 0, F1 = 1 (2)

The first numbers of this sequence are listed below

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377... (3)

These numbers can be generalized to obtain a different sequence of numberscalled the Lucas numbers which employ the same relation [6, 7]

Ln = Ln−1 + Ln−2 (4)

With seed values ofL0 = 0, L1 = 1 (5)

And are found in the integer sequence below

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843... (6)

Following the dynamic of the Pascal triangle, the Fibonacci numbers and ofthe Lucas numbers, two new triangles were created with different additionarrangements, one with the Fibonacci sequence and another with the Lucassequence. The goal was to see what sequences of numbers would come outof such arrangements, if any relations could be observed out of these and tobetter represent the Fibonacci relation and its generalizations such as theLucas sequence in the conditions of the Pascal triangle variation.

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2 The basics of the Pascal triangle and of its

variation

The principle of seed number and recurrence relation along with the under-standing of the Pascal triangle variation rules and addition arrangement iscritical to the good understanding of the results. These factors are closelyrelated to one another and their individual or common modification can altereach other, resulting in interesting or unwanted results.

2.1 Pascal triangle variation rules and addition arrange-ment

Unlike the original version of the Pascal triangle where one number was thesum of the above left and right numbers, the variation presents a differentapproach to the shape medium of the triangle. First of all the Fibonaccinumbers are placed at the left of the triangle in increasing order so that theapex of the triangle is equal to F0 as depicted in Figure 1

Then, the triangle is completed such as depicted in figure 1 and 2 for the vari-ation of Fibonacci and Lucas respectively. This process is repeated through-out the shape. The number sequence can be changed with any generalizationof the Fibonacci sequence as long as the recurrence relation (1) or (4) arerespected for most of the results below to remain true.

The Pascal triangle variation that is found reveals interesting properties thatwill be further explained in the result section of this paper

2.2 Fibonacci and Lucas seed numbers

The Fibonacci series use the seed numbers F0 = 0 and F1 = 1 to initiate thewhole sequence (3). If any of these numbers were to change, the sequence islikely to become very different. It is indeed what Edouard Anatole Lucas didby changing the seeds to F0 = 2 and F1 = 1 and gave what is now known asthe Lucas sequence (6).

For example, if the seed numbers of the recurrence relation Gn = Gn−1+Gn−2

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Figure 1: Pascal triangle variation rules with the Fibonacci sequence

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Figure 2: Pascal triangle variation rules with the Lucas sequence

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were to be G0 = 1 and G1 = 1 the following integer sequence would be found

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610... (7)

And so,Gn = Fn+1 (8)

For Fn having the interger sequence depicted in (3).

2.3 Fibonacci and Lucas recurrent relation

Although the different seeds numbers of the Fibonacci and Lucas sequencesproduce some quite different number suites, their recurrent relation is thesame. What it does is adding the two numbers to the left of the desirednumber starting with the sum of the given seeds.

It is also possible to change the recurrence relation in order to provide a dif-ferent number sequence that differs from the Fibonacci and Lucas sequences.For example, the recurrence relation

Hn = Hn−1 −Hn−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,−89... (10)

2.4 Bisection of a number sequence

The bisection of a number sequence only contain one part of the original num-ber sequence. The Fibonacci and Lucas number sequences are respectivelydefined such as

F2∗n;F2∗n−1 (11)

andL2∗n;L2∗n−1 (12)

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3 Results

While completing the Pascal triangle variation and exploring its properties,newfound number sequences and the their attributes, many interesting resultswere found and are presented below.

3.1 Pascal triangle variation properties

One of the most apparent properties about this variation is present as long asthe recurrent relations (1) or (4) are respected. In this sense, both Fibonaccian Lucas sequence will express the same properties in this subsection. Whenthe addition arrangement is respected, the Fibonacci sequence or any gener-alization of its seeds repeats itself to give the number sequences depicted inFigure 3.

Figure 3: Pascal triangle variation property with the Fibonacci sequence

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The the sum of each number present in individual latitudes of the trianglecan be expressed in the form below

n∑i=1

F2i−1 −n∑

i=1

Fi−1 (13)

Following the principles of this relation, it can be said that every number inthe Pascal triangle variation is part of the number sequence that initiated thetriangle since adding to consecutive numbers in these sequences will give thenext number. The geometry of the triangle makes this possible even thoughit 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 ofthe Fibonacci or Lucas sequences, depending on what sequence was used inthe making of the triangle, can be found repetitively like depicted in Figure4 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, 5778... (14)

1, 4, 11, 29, 76, 199, 521, 1364, 3571, 9349... (15)

For the Fibonacci sequence, the following bisections can be found

0, 1, 3, 8, 21, 55, 144, 377, 987, 2584... (16)

1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181... (17)

3.3 Pascal triangle variation sequence conjecture

An important conjecture is that numbers repeat themselves in the trianglein straight diagonal lines such as depicted in Figure 5 and the order of theselines respects the one of the initial number sequence. The sum of each ofthese lines could be expressed in the following if n is an even number andthe recurrent relation is the same as (1) and (4)

(n

2+ 1) · Fn; (

n

2+ 1) · Ln (18)

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Figure 4: Pascal triangle variation with the Fibonacci bisection sequences

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And if n is an odd number

(n

2+ 0.5) · Fn; (

n

2+ 0.5) · Ln (19)

If the initial sequence used in the making of the triangle is the Fibonaccisequence, this number sequence will result

0, 1, 2, 4, 9, 15, 32, 52, 105, 170... (20)

And for the Lucas sequence

2, 1, 6, 8, 21, 33, 72, 116, 235, 380... (21)

These sequences may differ significantly from their original number sequencesbut are closely related to them. The Pascal triangle variation tremendouslyhelp to show such relations.

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Figure 5: Pascal triangle variation relation with the Fibonacci sequence

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4 Conclusion

Starting from 3 classic concepts of discrete mathematics including the PascalTriangle, the Fibonacci sequence and the Lucas sequence, a new approachwas adopted to further investigate the variation of the well known Pascaltriangle while directly including the Fibonacci series and one of it’s mostpopular generalizations, the Lucas series.

After the variation was completed, it showed relevant properties absent fromthe original triangle. The first one of these was the presence of the self-repetition of the sequence used to complete the variation present in a partialform defined in the result section (11). This lead to the conclusion that thisrepresentation was of great value to better understand the workings of theFibonacci sequence recurrent relation Fn = Fn−1 + Fn−2. It also lead tothe 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 sequencesalready covered by other papers such as the Fibonacci and Lucas Bisectionnumber sequences which only included one half of their original Fibonacciand Lucas series.

Also, newfound sequences were explored such as the 3.3 sequence given bystraight lines featuring the same Fibonacci of Lucas numbers. The trianglehelped to show how much the differing sequences obtained were related tothe original sequence used in the making of the triangle.

Before interesting number series were found, the search for relations withinthem proved to be very difficult. The triangle might not have a direct ap-plications but it helps to show many mathematical concepts. Since only afew sequences and relations have been studied, it would be greatly interest-ing to investigate the several number series that have yet to be studied andthe possible relations they may have. Also, the geometry of the numbers inthe triangle remain of great interest and could have a potential for futureresearches.

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5 Acknowledgements

I would like to thank Julien Chartrand and Emile Boily-Auclair for theirguidance, help and moral support throughout the realization of this docu-ment.

References

[1] Peter Fox (1998). Cambridge University Library: the great collections.Cambridge University Press. p. 13. ISBN 978-0-521-62647-7.

[2] A. W. F. Edwards. Pascal’s arithmetical triangle: the story of a mathe-matical idea. JHU Press, 2002. Pages 3031.

[3] Gale, Thomson. ”Research Article: Pascal’s Triangle.” BookRags.BookRags, n.d. Web. 17 Feb. 2013.

[4] Goonatilake, Susantha (1998). Toward a Global Science. Indiana Univer-sity Press. p. 126. ISBN 978-0-253-33388-9.

[5] Singh, Parmanand (1985). ”The So-called Fibonacci numbers in an-cient and medieval India”. Historia Mathematica 12 (3): 22944.doi:10.1016/0315-0860(85)90021-7.

[6] Goonatilake, Susantha (1998). Toward a Global Science. Indiana Univer-sity Press. p. 126. ISBN 978-0-253-33388-9.

[7] Chris Caldwell, ”The Prime Glossary: Lucas prime” from The PrimePages.

[8] ”The Fibonacci Series - Biographies - Leonardo Fibonacci (ca.1175 -ca.1240)”. Library.thinkquest.org. Retrieved 2010-08-02.

[9] Leonardo Pisano - page 3: ”Contributions to number theory”. Encyclop-dia Britannica Online, 2006. Retrieved 18 September 2006.

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