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British Journal of Science
September 2014, Vol. 11 (2)
62
THE EXISTENCE OF FIBONACCI NUMBERS IN THE
ALGORITHMIC GENERATOR FOR COMBINATORIC PASCAL
TRIANGLE
BY
Amannah, Constance Izuchukwu
[email protected]; +234 8037720614
Department of Computer Science, Faculty of Natural and Applied Sciences,
Ignatius Ajuru University of Education, P.M.B. 5047, Port Harcourt, Rivers State,
Nigeria.
&
Nanwin, Nuka Domaka
Department of Computer Science, Faculty of Natural and Applied Sciences,
Ignatius Ajuru University of Education, P.M.B. 5047, Port Harcourt, Rivers State,
Nigeria
© 2014 British Journals ISSN 2047-3745
British Journal of Science
September 2014, Vol. 11 (2)
63
ABSTRACT
The discoveries of Leonard of Pisa, better known as Fibonacci, are
revolutionary contributions to the mathematical world. His best-known work
is the Fibonacci sequence, in which each new number is the sum of the two
numbers preceding it. When various operations and manipulations are
performed on the numbers of this sequence, beautiful and incredible patterns
begin to emerge. The numbers from this sequence are manifested throughout
nature in the forms and designs of many plants and animals and have also
been reproduced in various manners in art, architecture, and music. This
work simulated the Pascal triangle generator to produce the Fibonacci
numbers or sequence. The Fibonacci numbers are generated by simply taken
the sums of the "shallow" diagonals (shown in red) of Pascal's triangle. The
Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's
triangle. This Pascal triangle generator is a combinatoric algorithm that
outlines the steps necessary for generating the elements and their positions in
the rows of a Pascal triangle. The Pascal triangle generator is symbolized
with Eij. The is denote the element of a row while the js represent the
respective positions of the elements. The generated Fibonacci sequence from
the Eij model can be used in the following way; in the computational runtime analysis of Euclid's algorithm to determine the greatest common divisor
of two integers- the worst case input for this algorithm is a pair of
consecutive Fibonacci numbers; as pseudorandom number generators; The
Fibonacci numbers are also an example of a complete sequence. This means
that every positive integer can be written as a sum of Fibonacci numbers,
where any one number is used once at most. This work succeeded in
simulating the Pascal triangle to produce 20 Fibonacci numbers namely;
0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765.
KEY WORDS: Algorithm, Generator, Triangle, Fibonacci Numbers,
Sequence, Shallow Diagonal
INTRODUCTION
The discoveries of Leonard of Pisa, better known as Fibonacci, are
revolutionary contributions to the mathematical world. His best-known work is
the Fibonacci sequence, in which each new number is the sum of the two
numbers preceding it. When various operations and manipulations are
performed on the numbers of this sequence, beautiful and incredible patterns
begin to emerge. The numbers from this sequence are manifested throughout
© 2014 British Journals ISSN 2047-3745
British Journal of Science
September 2014, Vol. 11 (2)
64
nature in the forms and designs of many plants and animals and have also been
reproduced in various manners in art, architecture, and music. The
mathematician Leonardo of Pisa, better known as Fibonacci, had a significant
impact on mathematics. His contributions to mathematics have intrigued and
inspired people through the centuries
to
delve more deeply into the mathematical world. He is best known for the
sequence of numbers bearing his name.
Leonardo Pisano Bigollo (c. 1170 – c. 1250) [Wikipedia) – known as Fibonacci,
and also Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo
Fibonacci – was an Italian mathematician, considered by some "the most
talented western mathematician of the Middle Ages, Howard (1990). Fibonacci
is best known to the modern world for (Encyclopædia Britannica), the
spreading of the Hindu–Arabic numeral system in Europe, primarily through his
composition in 1202 of Liber Abaci (Book of Calculation), and for a number
sequence named the Fibonacci numbers after him, which he did not discover but
used as an example in the Liber Abaci, (Parmanand, 1986).
The Fibonacci sequence is named after Leonardo Fibonacci. His 1202 book
Liber Abaci introduced the sequence to Western European mathematics,
Goonatilake (1998), although the sequence had been described earlier in Indian
mathematics. (Knuth 2006; Singh 1985; Douady and Couder 1996). By modern
convention, the sequence begins either with F0 = 0 or with F1 = 1. The Liber
Abaci began the sequence with F1 = 1, without an initial 0.
Fibonacci numbers are closely related to Lucas numbers in that they are a
complementary pair of Lucas sequences. They are intimately connected with the
golden ratio; for example, the closest rational approximations to the ratio are
2/1, 3/2, 5/3, 8/5,.... Applications include computer algorithms such as the
Fibonacci search technique and the Fibonacci heap data structure, and graphs
called Fibonacci cubes used for interconnecting parallel and distributed systems.
They also appear in biological settings, Jones and Wilson (2006) such as
branching in trees, phyllotaxis (the arrangement of leaves on a stem), the fruit
sprouts of a pineapple, Brousseau (1969), the flowering of artichoke, an
uncurling fern and the arrangement of a pine cone. Knuth (2008). The Fibonacci
sequence appears in Indian mathematics, in connection with Sanskrit
prosody,(Singh 1985; Knuth 1968). In the Sanskrit oral tradition, there was
much emphasis on how long (L) syllables mix with the short (S), and counting
the different patterns of L and S within a given fixed length results in the
Fibonacci numbers; the number of patterns that are m short syllables long is the
Fibonacci number Fm + 1, (Knuth 2006).
© 2014 British Journals ISSN 2047-3745
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65
Liber Abaci also posed, and solved, a problem involving the growth of a
population of rabbits based on idealized assumptions. The solution, generation
by generation, was a sequence of numbers later known as Fibonacci numbers.
The number sequence was known to Indian mathematicians as early as the 6th
century Donald (2006) and Rachel (2008), but it was Fibonacci's Liber Abaci
that introduced it to the West.
In the Fibonacci sequence of numbers, each number is the sum of the previous
two numbers. Fibonacci began the sequence not with 0, 1, 1, 2, as modern
mathematicians do but with 1, 1, 2, etc. He carried the calculation up to the
thirteenth place (fourteenth in modern counting), that is 233, though another
manuscript carries it to the next place: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
233, 377.
The problem, dealing with the regeneration of rabbits, calculated the number of
rabbits after a year if there is only one pair the first month. The problem states
that it takes one month for a rabbit pair to mature, and the pair will then produce
one pair of rabbits each month following. Fibonacci’s solution stated that in the
first month there would be only one pair; the second month there would be one
adult pair and one baby pair; the third month there would be two adult pairs and
one baby pair; and so forth (Posamentier and Lehmann, 2007). When the total
number of rabbits for each month is listed, one after the other, it generates the
sequence of numbers for which Fibonacci is most famous:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377…
This string of numbers is known as the Fibonacci sequence, and each successive
term is found by adding the two preceding terms together. The Fibonacci
sequence is the oldest known recursive sequence, which is a sequence where
each successive term can only be found through performing operations on
previous terms. Interestingly, Fibonacci does not comment on the recursive
nature of this sequence. The relationship between the terms was not identified in
publication until four hundred years later. At the time of the publication of Liber
Abaci, no special notice was taken of these numbers. It was not until the mid1800s that mathematicians began to be intrigued by what would later be known
as the Fibonacci numbers (Posamentier and Lehmann, 2007). A closer
inspection of the numbers making up the Fibonacci sequence brings to light all
sorts of fascinating patterns and mathematical properties. Fibonacci himself
makes no mention of these patterns in his book, but the following patterns are a
few that have been brought to light over years of examination of the numbers in
the sequence. Any two consecutive Fibonacci numbers are relatively prime,
having no factors in common with each other (Garland, 1987). For example: 5,
8,13,21,34
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5 = 1 · 5; 8 = 2 · 2 · 2; 13 = 1 · 13; 21 = 3 · 7; 34 = 2 · 17
Summing together any ten consecutive Fibonacci numbers will always result in
a number which is divisible by eleven (Posamentier and Lehmann, 2007).
1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143
143/11 = 13
89 + 144 + 233 + 377 + 610 + 987 + 1,597 + 2,584 + 4,181 + 6,675 = 17,567
17,567/11 = 1,597
Following tradition, Fn will be used to represent the n-th Fibonacci number in
the sequence.
n
Fn
1
1
2
1
3
2
4
3
5
5
6
8
7
13
8
21
9
34
10
55
11
89
12
144
13
233
14
377
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15
67
610
Every third Fibonacci number is divisible by two, or F3. Every fourth Fibonacci
number is divisible by three, or F4. Every fifth Fibonacci number is divisible by
five, or F5. Every sixth Fibonacci number is divisible by eight, or F6, and the
pattern continues. In general, every nth Fibonacci number is divisible by the nth
number in the Fibonacci sequence, or FMN is divisible by FN (Garland, 1987).
Fibonacci numbers in composite-number positions are always composite
numbers, with the exception of the fourth Fibonacci number. In other words if n
is not a prime, the nth Fibonacci number will not be a prime (Posamentier and
Lehmann, 2007)
35).
F6 = 8
F9 = 34
F16 = 987
The reciprocal of the eleventh Fibonacci number, 89, can be found by adding
the Fibonacci sequence in such a fashion that each Fibonacci number
contributes one digit to the repeating decimal of the reciprocal, 1/89 (Garland,
1987).
0.0112358
13
21
34
55
89
144
233
377
610
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987
1/89 = 0.01123595505617787
Multiplying any Fibonacci number by two and subtracting the next number in
the sequence will result in the answer being the number two places before the
original (Garland, 1987).
…3,5,8,13,21,34,55,89,144,233…
2 · F6 –F7 = (2 · 8) - 13 = 16 - 13 = 3 = F4
2 · F11- F12= (2 · 89) - 144 = 178 - 144 = 34 – F9
2 · Fn – Fn+1= Fn-2
Summing consecutive odd-positioned Fibonacci numbers, starting with the first
odd-positioned number, F1, will result in a number that is the next Fibonacci
number in the sequence after the last term in the sum (Posamentier and
Lehmann, 2007).
F1 + F3 = 1 + 2 = 3 = F4
F1 + F3 + F5 = 1 + 2 + 5 = 8 = F6
F1+ F3+ F5 +F7 = 1 + 2 + 5 + 13 = 21 = F8
A similar pattern emerges when summing consecutive, even-positioned
Fibonacci numbers beginning with F2, only this time, the result is a number that
is one less than the Fibonacci number following the last even number in the sum
(Posamentier and Lehmann, 2007).
F2 + F4 = 1 + 3 = 4 = F5 - 1
F2 + F4 + F6 = 1 + 3 + 8 = 12 = F7 - 1
F2 + F4 + F6 + F8 = 1+ 3 + 8 + 21 = 33 = F9 - 1
The product of any Fibonacci number multiplied by the number two places after
it will be one more or one less than the square of the Fibonacci number between
the two. When the number to be squared is an even-positioned Fibonacci
number, one is added, and when it is odd-positioned, one is subtracted
(Posamentier & Lehmann, 2007).
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…3,5,8,13,21,34,55,89…
F4 · F6 = 3 · 8 = 24 and F25= 52= 25
F9 · F11= 34 · 89 = 3,026 and F210= 552 = 3,025
Pascal triangle on the other hand, measures trend of proportion in probability,
where it is used to find combinations. There are a number of principles that are
employed in solving combiantoric problems. These mathematical principles
include the Pigeon Hole Principle, the Stocks in Drawer Principle, the Counting
Principle, the Multiplication Principle, the Permutation Principle, the
Combination Principle and the Scheduling Principle.
Interesting, as these principles have proved to be, in their various numerical
results, the demands for a more technical and mathematically admissible
algorithm for an efficient generation of a scope free, nth-term Pascal triangle
remains in the dream of scientific computing. A synchronizing study of the
aforementioned combinatoric principles, undisputably translates this scientific
expectation to computing reality. The Pascal Generating algorithm in paper
provides a computing and mathematical model for achieving an nth term Pascal
Triangle. The Lemma, Pascal rule was introduced into the West by Blaise
Pascal, Riordan (1958).
The goal is to provide a unified treatment of analysis in combinatorics. Verlag’s
(2002) analysis as the core of the theory focuses on rational and meromorphic
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functions as well as two of their explanations given by verlag on fundamentals
of singularity analysis and combinatorial consequences, Stanley (1997; 1999).
1.1
Applications of Fibonacci Numbers
 The Fibonacci numbers are important in the computational run-time
analysis of Euclid's algorithm to determine the greatest common divisor
of two integers: the worst case input for this algorithm is a pair of
consecutive Fibonacci numbers, Knuth (1997).
 Yuri Matiyasevich was able to show that the Fibonacci numbers can be
defined by a Diophantine equation, which led to his original solution of
Hilbert's tenth problem.
 The Fibonacci numbers are also an example of a complete sequence. This
means that every positive integer can be written as a sum of Fibonacci
numbers, where any one number is used once at most.
 Moreover, every positive integer can be written in a unique way as the
sum of one or more distinct Fibonacci numbers in such a way that the
sum does not include any two consecutive Fibonacci numbers. This is
known as Zeckendorf's theorem, and a sum of Fibonacci numbers that
satisfies these conditions is called a Zeckendorf representation. The
Zeckendorf representation of a number can be used to derive its
Fibonacci coding.
 Fibonacci numbers are used by some pseudorandom number generators.
 Fibonacci numbers are used in a polyphase version of the merge sort
algorithm in which an unsorted list is divided into two lists whose lengths
correspond to sequential Fibonacci numbers – by dividing the list so that
the two parts have lengths in the approximate proportion φ. A tape-drive
implementation of the polyphase merge sort was described in The Art of
Computer Programming.
 Fibonacci numbers arise in the analysis of the Fibonacci heap data
structure.
 The Fibonacci cube is an undirected graph with a Fibonacci number of
nodes that has been proposed as a network topology for parallel
computing.
 A one-dimensional optimization method, called the Fibonacci search
technique, uses Fibonacci numbers.
2 METHODOLOGY
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The research design adopted in this work is simulation research. It simulated the
Pascal triangle generator to demonstrate the possibility of the Fibonacci
numbers. In the Pascal triangle we could see the incidence of the Fibonacci
numbers as much as we could see the angle of incidence in a prison experiment
in Physics. The direction of incidence of the Fibonacci numbers is along the
shallow diagonals of the Pascal triangle.
3 THE ALGORITMIC GENERATOR FOR THE COMBINATORIC
PASCAL TRIANGLE
3.1 The Lemma of Blaise Pascal
The formula for counting the number of ways to select m elements from a set
of n total elements can be characterized by this shorthand notation.
n
  
m
n!
m!(n  m)!
-
-
-
(1)
With this prelude, we can proceed to look at Pascal’s Triangle. The triangle
builds as follows.
(n=0)
1
(n=1)
1
(n=2)
1
(n=3)
1 3
1
2
3
1
1
In each case, the number in the next row is equal to the sum of the two numbers
directly above it.
Let us deduce certain principle that will enable us in preparing our model. Let
each item in a Pascal triangle be called element, denoted, by E. Let also all
elements in Row 1 be denoted by E1 and for elements in row 2, E3 for Row3,
© 2014 British Journals ISSN 2047-3745
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and Ei for Rowi for all i = 1 to nth row. The next step will be to represent the
position of each element given any nth number of rows, let Ej denote the
elements positions in the rows for all j = l to nth elements. E1 = First position
element, E2 = second position element, E 3 = third position element of the row
in question.
coronary 1:
Because the Pascal triangle is symmetric, E1 (clockwise) = E1 (anti-clockwise).
The centre of the row is the negotiation limit of the triangle. At this point, we
decide where to move, either to the left or to the right. All E j’s from the NL to
the left = All Ej’s from the NL to the right.
Synthesizing the algorithm we have
Eicjc = elements in rows is in positions js
Vi =
1, 2, 3 … nth element and
Vj =
1, 2, 3 . . . nth position
c = NL = Negotiating limit
Let’s see how this maps into the equation (a + b) n
(a + b)0 =
(1) 1
(a + b)1 =
(1) a
(a+b)2
=
(1)a2
(a+b)3
=
(1)a3 + (3)a2b + (3)ab2 + (1)b3
(a+b)4
=
(1)a4 + (4)a3b+ (6)a2b2 + (4) ab3 + (1)b4
+ (1)
b
(2) ab + (1) b2
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73
If we remember that;
0! = 1, we get the following;
For n = 0, the single element is equal to;
0
 
0

0!
(0  0)!
0!
=
1
3
3
3
1
1
For n = 1,    1,    1 For n = 2,    1,    3,    1
0
1
0
1
3
This is useful because it allows us to pick any exponent n and generate the list
of coefficients that makes up its values.
3.2
Proof of Pascal Rule
n
 n 1
 n 1 
  
  

m
 m 
 m 1
-
-
-
(2)
 n 1

 
 m 
(n  1)!
m![( n  1)  m]!
(n  1)!
m!(n  m  1)!
-
-
-
(3)
 n 1 

 
 m 1
(n  1)!
(m  1)! [( n  1)  (m  1)]!
-
-
-
(4)
=
(n  1)!

| (m  1)! (n  1  m  1)!
(n  1)!
(m  1)!(n  m)!
The trick here is to add both of fractional values.
M! = (m) (m-1)!
-
-
-
(5)
(n-m)! = (n-m) (n-m-1)!
-
-
-
(6)
Applying (5) and (6) to (2) and (3) we have
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September 2014, Vol. 11 (2)
 n 1
 n 1 

  
 
 m 
 m 1
=
[( n  m)  m]( n  1)!

m!(n  m)!
74
(n  m)( n  1)! (m)( n  1)!

m!(n  m)!
m!(n  m)!
n!
m1(n  m)!
-
-
-
(7)
-
(8)
The resolution shows that equations (1) and (7) are equal.
n
m
 n 1
 n 1 
 
  
 
 m 
 m 1
Row 1: =
n!
m!(n  m)!
-
-
E1
Row 2:
=
E11, E11
Row 3:
=
E21 + E22, E11 or E11, E21 + E22
Row 4:
=
E11, E31, + E32
Row 5:
=
E11, E41 + E42, E42 + E+43
Row 6:
=
E11, E51 + E52, E52 + E53
Row 7:
=
E11, E61 + E62, E62, E63, E63+ E64
Row 8:
=
E11, E71, + E72, E72, E73, E73+ E74, E74 + E75
Row 9:
=
E11, E81 + E82, E82 + E83, E83 + E84, E84, E85, E85
E86
Row 10:
=
E11, E91 + E92, E92+ E93, E93 + E94, E94 + E95,
E95 + E96, E96 + E97
Row 11:
=
E11, E101 + E102, E102 + E103, E103 + E104, E104 + E105, E105 + E106,
E106 + E107, E107 + E108
Row 12:
=
E11, E111 + E112, E112 +E113, E113 +E114, E114+E115, E105 + E116,
E116 + E117, E117 + E118,E118+ E119
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Row 13:
=
Row 14:
=
Row 15:
=
Row 16:
=
75
E11, E121+ E122, E122 +E123, E123 +E124, E124+E125,
E125 + E126,
E11, E131+ E132, E132 +E133, E133 +E134, E134
+E135,E135 + E136, E136 + E137, E137 + E138,E138 +
E139, E139+ E1310, E1310 + E1311
E11, E141+ E142, E142 +E143, E143 +E144, E144 +E145, E145 + E146,
E146 + E147, E147 +E126 + E127, E127 + E128,E128+ E129, E129 +E1210
E11, E151 + E152, E152 +E153, E153 +E154, E154 +E155, E155 + E156,
E156 + E157, E157 + E158, E158 + E159, E159+ E1510, E1510 +
E1511, E1511 + E1512, E1512 +E1513.
Rowij :
E11, Ei-11 + Ei-1 E1+ 1, Ei-1 E1+1 +Ei-1E1+2 + . . . Ei-1C
=
Fig.1: The Pascal Triangle Algorithmic Generator
1
1
2
1
1
3
3
1
1
4
6
4
1
1
5
10
10
5
1
1
6
15
20
15
6
1
1
7
21
35
35
21
7
1
1
8
28
56
70
56
28
8
1
1
9
36
84
126
126
84
36
9
1
1
10
45
120
210
252
210
120
45
10
1
1
11
55
165
330
462
462
330
165
55
11
1
12
66
220
495
792
924
792
495
220
66
12
1
13
78
286
715
1287
1716
1716
1287
715
286
78
1
14
91
364
1001
2002
3003
3432
3003
2002
1001
364
91
1
15
105
455
1365
3003
5005
6435
6435
5005
3001
1365
455
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76
1
16
120
560
1820
4368
8008
11440
12870
11440
8008
4368
1820
560
1
17
136
680
2380
6188
12376
19448
24310
24310
19448
12376
6188
2380
680
1
18
153
816
3060
8568
18566
31724
43758
48620
43758
31724
31724
18566
8568
1
19
171
969
3876
11628
27134
40290
75482
82378
92378
75482
40290
27134
20
162
1140
4845
15504
38762
67424
115772
157860
175756
177860
115772
67424
38762
1
Fig. 2: Results of the Pascal Generator
The combinatoric algorithm represents half-symmetry, ½C of the Pascal
triangle. A complete symmetry is the collection of the Ei; of both right and left,
from NL or C, where C is the centre. It is of little or no essence to repeat ½C
from left of right or vice versa. So we terminate our algorithm at half symmetry
½C.
1. The Fibonacci Model
By definition, the first two numbers in the Fibonacci sequence are 1 and 1, or 0
and 1, depending on the chosen starting point of the sequence, and each
subsequent number is the sum of the previous two.
Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
or (often, in modern usage):
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181,
6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229,
832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352,
24157817, 39088169
In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the
recurrence relation
or
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4.1
77
List of Fibonacci Numbers
The first 21 Fibonacci numbers Fn for n = 0, 1, 2, ..., 20 are: Bona (2000)
F F F F F F F F F F F1 F1
0
1
2
3
4
5
6
0 1 1 2 3 5 8
7
8
9
0
1
F12 F13 F14 F15 F16 F17 F18 F19 F20
1 2 3
14 23 37 61 98 159 258 418 676
55 89
3 1 4
4 3 7 0 7
7
4
1
5
The sequence can also be extended to negative index n using the re-arranged
recurrence relation
which yields the sequence of "negafibonacci" numbers Lemmeryer (2000)
satisfying
Thus the bidirectional sequence is
F−8 F−7 F−6 F−5 F−4 F−3 F−2 F−1 F0 F1 F2 F3 F4 F5 F6 F7 F8
−21 13 −8 5 −3 2 −1 1 0 1 1 2 3 5 8 13 21
4.2
Existence of Fibonacci Numbers in Pascal Triangle
Fig. 3: Generating Fibonacci Numbers from Pascal Triangle
Figure 3 above contains Fibonacci numbers for n+10. The Fibonacci numbers,
fn =1,1,2,3,5,8,13,21,34,55,89
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Fi=1, f2=1, f3=2, f4=3, f5=5, f6=8, f7=13, f8=21, f9=34, f10=55
0
1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765
10946
17711
28657
46368
75025
0
1
1
1+1
1+2
2+3
3+5
5+8
8+13
13+21
21+34
34+55
55+89
89+144
144+233
233+377
377+610
610+987
987+1597
1597+2584
2584+4181
4181+6765
6765+10946
10946+17711
17711+28657
28657+46368
The Fibonacci numbers are the sums of the "shallow" diagonals (shown in red)
of Pascal's triangle. The Fibonacci numbers occur in the sums of "shallow"
diagonals in Pascal's triangle. Lucas (1891).
These numbers also give the solution to certain enumerative problems.
Pisano(2002). The most common of such problem is that of counting the
number of compositions of 1s and 2s that sum to a given total n: there are Fn+1
ways to do this. For example F6 = 8 counts the eight compositions:
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1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 =
1+2+2,
all of which sum to 6−1 = 5.
This configuration has many interesting and important properties:



Notice the left-right symmetry - it is its own mirror image.
Notice that in each row, the second number counts the row.
Notice that in each row, the 2nd + the 3rd counts the number of numbers
above that line.
There are endless variations on this theme.
Next, notice what happens when we add up the numbers in each row - we get
our doubling sequence.
Now for visual convenience we draw the triangle left-justified. Add up the
numbers on the various diagonals...
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and we get 1, 1, 2, 3, 5, 8, 13, . . . the Fibonacci sequence!
4.3
Applications of Fibonacci Numbers
 Fibonacci numbers are used by some pseudorandom number generators.
 Fibonacci numbers are used in a polyphase version of the merge sort
algorithm in which an unsorted list is divided into two lists whose lengths
correspond to sequential Fibonacci numbers – by dividing the list so that
the two parts have lengths in the approximate proportion φ.
 Fibonacci numbers arise in the analysis of the Fibonacci heap data
structure.
 The Fibonacci cube is an undirected graph with a Fibonacci number of
nodes that has been proposed as a network topology for parallel
computing.
 A one-dimensional optimization method, called the Fibonacci search
technique, uses Fibonacci numbers, Knuth (1997).
4.4
Fibonacci primes
A Fibonacci prime is a Fibonacci number that is prime. The first few are:
2, 3, 5, 13, 89, 233, 1597, 28657, 514229, …
Fibonacci primes with thousands of digits have been found, but it is not known
whether there are infinitely many.
5.
RESULTS AND DISCUSSION
Figure 4 below contains the first 20 Fibonacci numbers from the generated
Pascal triangle. This is the result of simple summation of the shallow diagonal
the Pascal triangle.
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Fig. 4: Resultant Fibonacci Numbers from the Pascal Triangle
5. CONCLUSION
This unique and fascinating string of numbers possesses all sorts of intriguing
properties, which can be discovered by applying various mathematical
procedures to the numbers in the sequence. Fibonacci numbers are present
throughout the world in which we live, and the patterns which can be formed
from them both astonish and perplex the mind. The Fibonacci numbers are
beautiful to study in and of themselves, but there is a higher beauty to them as
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These numbers highlight the incredible order and mathematical complexity of
the world we live in, which all points to the Creator. Such intricacies could not
have evolved by mere chance, but are the work of a God of order, who created
all things.
Fibonacci numbers can be applied to an elementary example of geometric growth - asexual
reproduction, like that of the amoeba. Each organism splits into two after an interval of
maturation time characteristic of the species. This interval varies randomly but within a
certain range according to external conditions, like temperature, availability of nutrients and
so on. We can imagine a simplified model where, under perfect conditions, all amoebae split
after the same time period of growth. So, one amoeba becomes two, two become 4, then 8,
16, 32, and so on.
Fig. 5: Doubling Sequence in asexual Reproduction in Amoeba
We get a doubling sequence. Notice the recursive formula:

An =2An
This of course leads to exponential growth, one characteristic pattern of population growth.
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