Running Head: FIBONACCI AND LUCAS NUMBERS
The Infinite Reaches of Fibonacci and Lucas Numbers
David Musser
University of Kentucky
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Running Head: FIBONACCI AND LUCAS NUMBERS
There are many aspects of mathematics that are truly astounding and perhaps sometimes
unexplainable. Truths that seemingly exist without any apparent reason or cause yet somehow
help us make sense of the world around us. One man, named Leonardo Bonacci, used the
breeding pattern of rabbits to create a sequence that consequently unlocked the secret of the
golden ratio (often denoted as
). This in turn is intimately connected to man’s perception of
beauty, art, music, and architecture - even the growth of a single daisy to the development of the
vast and infinite expanse of a galaxy. How could these things from such differing realms
possibly be connected? The answer is in the Fibonacci sequence and the closely related Lucas
numbers. Through the study of these sequence one can begin to understand the magic that is
mathematics as well as the beauty of its simplicity - even though the full magnitude of its
implications can never be entirely grasped.
History
Leonardo Bonacci was born sometime around the year 1170 in Pisa in what is now
known as Italy. He was born the son of a merchant and traveled extensively with his father, who
directed a trading post in Bugia, a port city in the Almohad Dynasty in present day Algeria.
Through these travels, Leonardo, commonly known as Fibonacci, would become familiar with
the Hindu-Arabic numeral system (Wikipedia, 2014).
Recognizing that arithmetic manipulation using this system was easier and more effective
than using the Roman numerals system, Fibonacci traveled all over the Mediterranean world
studying under the leading Arab mathematicians of his day. He returned to Pisa at the age of 30
and for the next two years worked on what would become his greatest work, his Book of
Calculation, the Liber Abaci. (Wikipedia, 2014). This extensive piece contained nearly all of
the mathematical and algebraic knowledge of the time.
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Running Head: FIBONACCI AND LUCAS NUMBERS
With the publication of Liber Abaci in 1202 AD, Europe began to see a spread in the use
of the Hindu-Arabic system. One of the main ideas of this book that had lasting implications
was that it advocated the use of numerals 0-9. He began chapter 1 with:
The nine Indian figures are: 9 8 7 6 5 4 3 2 1.
With these nine figures, and with the sign 0, which the Arabs call zephir, any
number whatsoever is written (Sigler, 2002, 1).
This then was the first formal representation of the base-10 system in the Western world that we
use today (Posamentier, 2007, 11).
With his background at the trade post, he used specific examples as they related to
commercial bookkeeping, conversion of weights and measures, interest calculations, and money
changing to support his claims of the superiority of the Hindu-Arabic system. In the well
educated Mediterranean world, this book was well received, largely accepted, and had a
profound effect on modern thought of that period.
In this book, he posed the following question:
A certain man had one pair of rabbits together in a certain enclosed place, and one
wishes to know how many are created from the pair in one year when it is the nature
of them in a single month to bear another pair, and in the second month those born to
bear also. Because the abovewritten pair in the first month bore, you will double it;
there will be two pairs in one month. One of these, namely the first, bears in the
second month and thus there are in the second month 3 pairs; of these in one month
two are pregnant, and in the third month 2 pairs of rabbits are born, and thus there are
5 pairs in the month; in this month 3 pairs are pregnant, and in the fourth month there
are 8 pairs, of which 5 pairs bear another 5 pairs; these are added to the 8 pairs
making 13 pairs in a fifth month; these 5 pairs that are born in this month do not mate
in this month, but another 8 pairs are pregnant, and thus there are in the sixth month
21 pairs;....
The answer to this problem, as discovered by Fibonacci, is what is known today as the
Fibonacci Sequence. A sequence is simply an ordered set of quantities. The following are all
sequences: 1, 2, 3, 4, 5…; 2, 4, 6, 8, 10…; 2, 4, 8, 16, 32…; If Fibonacci had not specified a
month for the newborn pair to mature, he would not have a sequence named after him because
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Running Head: FIBONACCI AND LUCAS NUMBERS
the number of pairs would simply double each month. Where after n months there would be 2𝑛
rabbits. This is plenty of rabbits but not mathematically a distinguished concept. Because
Fibonacci’s approach relies on the ordering of the month, then the key is that the number of
rabbits at the end of a month is the number at the beginning of the month plus the number of
births produced by the mature pairs (Moler, 2011, Ch 2).
Although the sequence as identified in the question about rabbits is named after him, it
had been described much earlier in oriental mathematics, specifically in ancient and medieval
India. Virahanka is attributed with the discovery of this connection as early as 600 AD in his
analysis on variations of poetic meter, or morae, as captured in his early work
Vrttajatisamuccaya. His translation (Sanskrit) of the poetic rule (Prakrit) states, “the variations
of two earlier meters being mixed, the number is obtained. That is a direction for knowing a
number [of variations] of the next matra(-vrtta)” (Singh, 1985, 229-233). Matra-vratta are
meters in which the number of morae remains constant while the number of letters is arbitrary which is a type of meter in Sanskrit and Prakrit poetry, common to that era.
Gopala was an author who wrote a commentary on a palm leaf manuscript of Virahanka’s
rule of the numbers of variation. This manuscript was handwritten sometime around the year
1134 AD and goes into much greater detail on Virahanka’s rule:
Variations of two earlier meters [is a variation] of matra-vratta.
For example, for [a meter of] three [morae], variations of two earlier meters, one and
two being mixed three happens.
For [a meter] of four [morae], variations of meters of two morae [and] three morae
being mixed, five happens.
For [a meter] of five [morae] variations of two earlier [meters] of three morae [and]
of four morae, being mixed, eight is obtained.
In this way, for [a meter] of six morae, [variations] of four morae [and] of five morae
being mixed, thirteen happens. And like that, variations of two earlier meters being
mixed, [variations of a meter] of seven morae [is] twenty-one.
In this way, the process should be followed in all matra-vrttas. (Velankar 1962, 101)
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Running Head: FIBONACCI AND LUCAS NUMBERS
Therefore, this famed sequence has no doubt been around long before Fibonacci ever
published it in 1202 AD; however the spread of ideas was much more easily captured and
entrenched in the Roman-centric world in which he lived rather than the lesser known East of
modern day India. However, the Fibonacci numbers did not see their full prominence until the
French mathematician Edouard Lucas studied them in the late 1800s. He began his sequence
with a 1 and 3 rather than a 1 and 1. We will look at this more in depth later on.
The Definition
Using the question of the rabbits, Fibonacci determined that if the rabbits reached
maturity two months after birth and produced a new pair every month thereafter, then the number
of rabbits would increase monthly according to a sequence. This sequence states that 𝐹𝑛 (where
the subscript n is used to determine the specific iteration in the sequence such that
is the
16th Fibonacci number) satisfies the recurrence equation 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 where 𝐹1 = 1 and
𝐹2 = 1 (or often in modern usage, 𝐹0 = 0 and 𝐹1 = 1) to look like
1,1,2,3,5,8,13,21,34,55,89,144. .. (Wikipedia, 2014). This is the solution to the problem of the
rabbits beginning with 1 pair (or the first month with 2, second month with 3, third month with 5
and so on) (Posamentier, 2007, 11). The simple way to think about this sequence is that each
number is the sum of the preceding two numbers.
Mathematical Connections
The image to the right shows a pattern
denoted by Pascal found in successive triangular
numbers (pictured are the first 4 triangular numbers). This concept was later developed and
adapted into what is known today as Pascal’s Triangle (Edwards, 1987, 2). The Fibonacci
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Running Head: FIBONACCI AND LUCAS NUMBERS
numbers occur in the shallow diagonals of Pascal’s triangle as denoted below. Early Indian
mathematicians were able to deduce this same idea although they had different representations
for Fibonacci sequence as well as Pascal’s triangle.
They used this method to determine the number of
variations of a meter. However it also established a
relation between the sequence of binomial
coefficients and the Fibonacci numbers (Singh, 1985,
236-237). The similarities to the aforementioned
Pascal’s Triangle are quite evident although denoted in a slightly different representation. When
comparing the above triangle with the image to the right, one can see how each shallow diagonal
of the triangle is the same as each row above.
Indian mathematicians constructed this
pyramid structure by adding specific diagonals
to create the underlying cells. Notice also that
each pair begins on the left with 1 over 2, 1
over 3, 1 over 4… and ends with 1 over 1…; the second to last set maintains the pattern 1 over 2,
3 over 4, 5 over 6…; while the intermittent cells are created by the various diagonals such that
1+2=3, 3+3=6, 4+6=10...
These numbers also provide the solution to certain problems in enumerative
combinatorics, a branch of mathematics concerning the study of finite or countable discrete
structures. The most basic form of enumerative combinatorics is one that counts the number of
elements in a set that is made up of a combination of ones and twos. Fibonacci numbers provide
a framework for counting permutations, combinations, and partitions (Wikipedia, 2014). Take a
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Running Head: FIBONACCI AND LUCAS NUMBERS
look at the Fibonacci number
, which produces
: which produces 13 compositions that equal
, or 4 combinations; as well as
, or 6, where n is the specified
iteration of the sequence.
This same pattern exists for all
. The Fibonacci sequence is riddled with mysterious patterns
(Benjamin, 2013).
One of the famous methods of representing the sequence can be found in the summation
of squares. (Notice in the image on the following page that you continue to build rectangles
when looking at the whole) Take the given numbers
and square them to get
Notice that adding any given two consecutive squares will result in a Fibonacci number of the
next odd iteration such that
If we simply add all the squares note that
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Running Head: FIBONACCI AND LUCAS NUMBERS
While these sums are not themselves numbers in the Fibonacci sequence - take a look at how
they connect:
These sums are thus consecutive iterations in the sequence multiplied together (Benjamin, 2013).
This can easily be seen when looking at the geometry of
the figure to the left. Take a look at the three squares
1,1,2 that combined make a rectangle of
where
. The four squares 1,1,2,3 make a rectangle of
; the five squares 1,1,2,3,5 make a rectangle
of
and so on. Essentially what we have
done in adding the squared numbers of the sequence can easily be thought of in terms of area of
the above shape. One way to think of it is that every consecutive pair of numbers in the
sequence is a rectangle of that same base and height. As in rectangles of: 2x3, 3x5, 5x8, 8x13,
13x21, 21x34… and so on.
As we continue to explore the numbers in this
sequence, let us dig into these proportions by dividing
the larger number by the smaller number. What we
discover here is that the Fibonacci sequence when taken
as consecutive quotients quickly begins to approximate
the golden ratio. Often denoted as
throughout history, the golden ratio is very simply defined
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Running Head: FIBONACCI AND LUCAS NUMBERS
as
(MathIsFun). The quotients of the consecutive numbers of the
sequence took like:
Notice that these numbers are consistently approximating
Consider then an even
larger number in the sequence- take the forty-first number over the fortieth:
(Posamentier, 2007, 13). Note how this number is almost identical to
One interesting thing about
, the golden number.
is that it is self-defining, or infinite. Notice the equation
. This would begin to take the form
, making it an irrational
number, or one that cannot be simply defined by any specific fraction. Any irrational number
can be expanded into an infinite continued simple fraction (Niven, 1956, 60). The correlation
with the Golden ratio and the Fibonacci sequence is truly fascinating. Some of the most
spectacular feats of art and architecture stem from this ratio (MathIsFun).
Fibonacci and the Universe
Let us examine some of the most sensational works of man. Arguably the most famous
piece of art, the Mona Lisa, is indeed bounded with this ratio
. The spiral is regarded as one of
the most beautiful ratios ever established. One can see on the image below what is known as the
Golden Spiral, or often the Fibonacci spiral. This is simply an approximation to the Golden
Spiral using quarter circle arcs that are inscribed in each square of the Fibonacci series. During
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Running Head: FIBONACCI AND LUCAS NUMBERS
the Renaissance, it was determined that harmony, balance, and beauty in art could best be
represented through particular numbers - especially in relationship to other numbers. Leonardo
da Vinci masterfully constructed the proportions of the human body based
on the Golden Section, which is the numerical breakdown of the Golden
Spiral; which is also the Fibonacci sequence (Posamentier,
2007, 75).
It would seem that the magic of Fibonacci exists even in
nature. This is a truly mind shattering phenomenon. How
can mathematics, a purely intangible, unnatural result of logic, reason,
and thought - exist in the natural and physical world around us? As it
turns out, plants grow new cells in spirals, both in the negative
(clockwise) and positive (counterclockwise) direction. In the example
below, there are 34 clockwise spirals and 21 counterclockwise spirals; both of which are
consecutive numbers in the Fibonacci sequence. Leaves, branches, and petals have all been
known to grow in spirals as well. This spiral can also be found in a common conch at the bottom
of the ocean, as well as the spirals of a pinecone, or even the unfurling of a fern. In biology, this
is known as phyllotaxis, or the arrangement of leaves. It turns out most plants have a number of
petals that correspond directly to Fibonacci numbers.
Phyllotaxis can be
determined by finding the lowest
leaf and following the spiral up the
branch until you locate the next leaf
that is directly over the initial leaf.
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Running Head: FIBONACCI AND LUCAS NUMBERS
The number of leaves that you pass along the way will always be a Fibonacci number. This
genetic phenomenon of nature is known as the “genetic spiral of a plant”. Note that the
phyllotaxis will be different for the different species. While this ratio is not guaranteed for every
plant, it can be observed on most plants (Posamentier, 2007, 72).
This ever-increasing mystery of the
Golden spiral is also seen in other areas of
nature like the formation of hurricanes, or
even galaxies (MathIsFun, 2014). From the
structure of the face, to the proportions
found between body segments, even all the
way to embryonic geometry, it seems
almost everything can relate to the Fibonacci sequnce! The very creation of a fetus displays the
harmonic interference pattern of the spiral and circle, the same Golden Ratio:
. We even see
this sequence in DNA molecules. Each full cycle of a double helix is 34 angstroms long by 21
angstroms wide (SpiralConspiracy). Where else could this sequence show itself?
The truth is, music has its very foundation built on the Fibonacci sequence. Let’s look at
how these numbers are incorporated into a simple octave.
Notice that an octave contains 13 chromatic (or consecutive)
notes, while a standard scale of any given key is 8 notes. There
are 5 black notes in an octave, divided into groups of 3 and 2.
A scale is a set of tones based on 2 steps or 1 step from the root tone or first note of the scale. In
any given chord, the 3rd and 5th notes compose the harmonies and the 8th is the octave
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Running Head: FIBONACCI AND LUCAS NUMBERS
(Meisner, 2012). Music is littered with Fibonacci numbers! Not only in terms of music theory,
but also in the science behind harmonics as they relate to pitch and wavelength.
The scales used in western music are built from the natural harmonics that are created by
ratios of frequencies. These ratios are taken directly out of the Fibonacci sequence. 1/1 is the
root ratio for a given note; typically the foundational note
is La, or A and has a frequency of 440 Hz. This is the
starting point when tuning a piano, or the note the first
chair violinist uses to tune an orchestra. All other notes
can be built out of this ratio, and use the Fibonacci
sequence to do so. Not only is this sequence used in the
creation of notes and pitches, it is also used in the
aesthetics of instrument design! (Meisner, 2012)
Connections can also be drawn from Mozart and
Beethoven symphonies and their correlation to the magic
of Fibonacci and
. These connections are noticed in the
specific measures that these composers used to start a new phrase, or return to an already used
melody. This mathematical and deliberate undertone created an incomparable balance of
structure that is intuitively pleasing to listeners of all ages. It is also largely speculated that
Bach, Satie, Debussy, Schubert, and Bartók also used golden numbers and the golden section in
their compositions. Many scholars have researched these components in relation to various
famed pieces by these composers and have discovered a good amount of supportive data that
indeed
was used thoughtfully in their creation (Knot, 2014).
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Running Head: FIBONACCI AND LUCAS NUMBERS
How is something so simple as
connected to so much of life and the
world around us? There are many doubters and skeptics out there who question the validity of
this sequence and attribute it to dumb luck or coincidence. However the fact is that it is naturally
occurring all too often to be blind chance. Another sequence exists in close relation to Fibonacci
numbers. These are known as the Lucas numbers.
Lucas Numbers
Edouard Lucas investigated patterns found within the Fibonacci sequence and discovered
a sequence with similar properties. His sequence used the same concept of adding the previous
two to get the next value, however Lucas started
his sequence with 2 and 1 and it looks like:
for
where
. Compare these numbers to the Fibonacci sequence and one will find a close
relation. If we add up alternating Fibonacci numbers, we will see the given Lucas number of the
skipped iteration. This follows the formula
for all integers of
where the
Lucas number 𝐿𝑛 is equal to any given two Fibonacci numbers of a skipped iteration such that:
(Check these figures on the table above, each one will work).
We can use a similar formula for adding alternate Lucas numbers. The formula is
for all integers . Notice the pattern,
the sum of the Lucas numbers such that:
13
is always 5 times the result of
Running Head: FIBONACCI AND LUCAS NUMBERS
Because their recursive definitions are the same, similar patterns can be discovered with
the
and so on. There always seems to be a correlation and connection between
the two sequences (Knot, 2014).
What is it within us as people that find such a deep connection to these numbers?
Another interesting aspect of these sequences is the fact that they have been used across different
cultures that existed independently in different contexts of history and geography. Yet
somehow, multiple groups of people have been able to figure out the same ratio along with its
inherently aesthetic value. This concept is perhaps the most baffling of them all.
Fibonacci in the Modern World
The vast connections between these sequences and the universe around us are truly
magnificent and astounding. They do indeed hold more value than simply being cool facts.
There are plenty of interesting applications for these numbers in the modern world. We
especially find uses in computer algorithms. The structured and deliberate nature of this
sequence makes it convenient to use in the creation of programs and graphics.
One way we see it used is in algorithms such as the Fibonacci search technique or the
Fibonacci heap data structure. Using this sequence as a search technique is a method that
searches any sorted array with a divide-and-conquer mentality. This narrows down the search
with the aid of the Fibonacci sequence. This search algorithm has a slight advantage over the
more common binary search because it uses up slightly less time trying to find the storage
location (Wikipedia, 2014).
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Running Head: FIBONACCI AND LUCAS NUMBERS
A Fibonacci heap is commonly used in computer science applications. This is a heap
data structure that consists of trees. It has a better running time than the binomial heap because
the Fibonacci numbers are used in the running time analysis. Essentially, the key of the child is
always greater than or equal to that of the parent. So the minimum key must always exist at the
root of one of the trees. The size of the subtree rooted in the node of the kth degree is at least
where
is the kth Fibonacci number (Wikipedia, 2014).
In numerical analysis, specifically when looking at
super linear convergence, the secant method for root-finding
has a convergence order of
. It can be thought of as a
finite difference approximation of Newton’s method
(Wikipedia, 2014). The secant method is a root-finding
algorithm that uses the succession of roots of secant lines to
better approximate a root of a function f. It is faster than the bisection method, however it is not
as fast as Newton’s method. It is however special in the fact that it does not require f to be
differentiable, or for f’ to exist.
This sequence is also used as a graph called Fibonacci cubes. This is used for
interconnecting distributed and parallel systems. These are mathematically similar to the
hypercube graphs, however
they have a Fibonacci
number of vertices. The
only labels allowed in a
Fibonacci cube of
order can be labeled with
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Running Head: FIBONACCI AND LUCAS NUMBERS
bitstrings with no two consecutive 1-bits. There are
vertices because there are
labels possible. These cubes are commonly used in chemical graphs to describe perfect
matchings of specific molecular graphs (Wikipedia, 2014).
There are truly fascinating connections with this linear recurrence equation. For
example, Lucas numbers
is the number of ways by which you pick a set (empty set included)
without picking two consecutive numbers. The only square numbers in the Lucas sequence are 1
and 4, while the only triangular numbers are 1, 3, and 5778. The only cubic number is 1.
Interestingly enough, if
true. Composite numbers
is prime,
. However the converse is not necessarily
such that
are known as Lucas pseudoprimes and
have this property. This congruence holds for every prime number in
(Weisstein, 2014).
The connection between fractals and the linear recurrence equation used by Fibonacci is
also quite profound. A fractal is simply a never-ending pattern. Perhaps the most famed fractal
design is the Mandelbrot set. The cool thing about
these designs is that you can zoom in on them an
infinite amount of times because they always repeat
the pattern. How is this connected to the Fibonacci
sequence?
Fractals and Fibonacci numbers are deeply rooted in the same concept and share many
similar properties. The way the veins of a leaf are divided along the body of the leaf share
inherently the same ideas as the infinite properties of fractals and Fibonacci numbers. These
rates can be applied to nature to build the correct exponential model for growth.
There seem to be hundreds of related recurrence equations as well as additional identities
that are satisfied by Lucas and Fibonacci numbers. It seems like one can apply these series’ to
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Running Head: FIBONACCI AND LUCAS NUMBERS
any other formula that already exists in mathematics. With Fibonacci holding the answer to so
many diverse areas of math and science, is there anything that we do not know about it?
Actually, one of the unsolved problems in mathematics is concerning the Fibonacci
sequence and primes. The question seems simple: are there an infinite number of Fibonacci
primes? The first few proven Fibonacci primes are: 2, 3, 5, 13, 89, 233, 1597, 28657, 514229,
433494437, 2971215073,... The largest known probable prime was recently discovered by Henri
Lifchitz in 2014. It is
and has 606,974 decimal digits (Wikipedia, 2014). This is
absolutely mind shattering to think about.
Primes have always been tricky, which is how we have things like cyber security through
encryption. But mixed with the predictability of the Fibonacci sequence - could this be the key
to unlocking the answer of the primes? The question has yet to be solved, though it is cause to
make one pause and ponder the implications.
The magic of these sequences is truly magnificent and seemingly abundant. What future
discoveries will be made with regards to connections in the Fibonacci sequence? In the study of
mathematics, we are always told to solve for x and we plug away at equations often mindlessly,
but it is important to think about y (why). There is undoubtedly a profound beauty to
mathematics that is easy to lose sight of in the midst of questing for the right answer. The
seemingly inexplicable beauty of Fibonacci is merely a step toward a greater appreciation of
mathematics. Appreciating mathematics for its intrinsic properties is not something that can be
inherently taught, but must be discovered on one’s own.
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Running Head: FIBONACCI AND LUCAS NUMBERS
References
Alfred, B. U. (1965). An Introduction to Fibonacci Discovery. The Fibonacci Association.
Benjamin, A. (2013, June). The Magic of Fibonacci Numbers. TED. Retrieved from
http://www.ted.com/talks/
Edwards, A. W. F. (1987). Pascal’s Arithmetical Triangle. Charles Griffin & Company Limited
Knot, R. (2014, April 28). The Lucas Numbers. Retrieved from http://www.maths.surrey.ac.uk/
MathIsFun. (2014) Golden Ratio. MathIsFun. Retrieved from http://www.mathsisfun.com/
Meisner, G. (2012, May 4). Music and the Fibonacci Sequence and Phi. The Golden Number.
Retrieved from http://www.goldennumber.net/
Moler, C. (2011). Experiments with MATLAB: Fibonacci numbers. E-book.
Niven, I. (1956). Irrational Numbers. The Mathematical Association of America.
Posamentier, A. S. (2007). The Fabulous Fibonacci Numbers. Prometheus Books.
Sigler, L. E. (2002). Fibonacci’s Liber Abaci: an English translation. Springer-Verlag New
York, Inc.
Singh, P. (1985). Historia Mathematica: the so-called Fibonacci numbers in ancient and
medieval India. (Vol. 12).
SpiralConspiracy. (2014) Fibonacci Sequence. Wordpress. Retrieved from
http://spiralconspiracy.wordpress.com/
Velankar, H. D. (1962). Vrttajatisamuccaya of Kavi Virahanka. Jodhpur: Rajasthan Oriental
Research Institute.
Wikipedia. (2014). Fibonacci Number. Wikipedia. Retrieved from http://en.wikipedia.org/
Weisstein, E. W. (2014). Lucas Number. WolframAlpha. Retrieved from
http://mathworld.wolfram.com/
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