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Fibonacci Sequences of Higher Orders
Fibonacci Sequences of Higher Orders: Sums, Differences, and Diagrams
Cole Jeznach
STEM Research Project
Massachusetts Academy of Mathematics and Science
February 24, 2014
Fibonacci Sequences of Higher Orders 1
Table of Contents
Abstract
2
Introduction
3
Literature Review
4
Mathematical Objective
13
Methodology
14
Results
17
Discussion
Conclusions
33
Limitations and Assumptions
34
Applications and Future Experiments
35
Literature Cited
36
Acknowledgements
38
Fibonacci Sequences of Higher Orders 2
Abstract
The Fibonacci sequence has been intensely studied for hundreds of years after its
appearance in the Liber Abaci by Leonardo Bigollo. The terms of the sequence have been found
in several branches of mathematics, botany, art, and architecture. The purpose of this investigation
was to analyze the sums of Fibonacci sequences with any arbitrary start terms and of any order
and to identify and prove the limit of converging ratios of Fibonacci sequences of higher order as
the order increases to infinity. Patterns of many data tables were identified and proven using proof
by induction and different mathematical theorems of discrete dynamical systems and of summation
formulas. It was proven that as the order of a Fibonacci sequence increases to infinity, the
converging ratio of consecutive terms has a limit of 2. It was also shown that the sum of Fibonacci
sequences of the same order yields another Fibonacci sequence of that order that begins with start
terms equal to the sum of the start terms of the original sequences being added. Consequently, the
proofs have suggested the ability to draw a Fibonacci tree diagram of any order.
Fibonacci Sequences of Higher Orders 3
Introduction
The most well-known Fibonacci sequence is the sequence that begins as 1, 1, 2, 3, 5, 8, 13,
21, 34. This sequence is a list of numbers in which the last two numbers are summed to yield the
next one, starting at the number one. Leonardo Pisano Bigollo created the list of numbers in his
famous book, Liber Abaci, in which he tried to explain the pattern of rabbit population growth. In
this math problem, one pair of rabbits could produce two newborn rabbits every month. However,
these newborns could only reproduce after a two-month inactivity period. The sum of all of the
pairs of rabbits at the end of each month formed the Fibonacci sequence (Moore, 2008).
A surge in interest in the sequence led to concentrated study in its hidden patterns and rules.
Given the Fibonacci sequence defined as Fn=Fn-1+Fn-2, where F0=0 and F1=1, many different
mathematical patterns can be proven. For example, the greatest common divisor of each number
and the next in the sequence (Fn and Fn+1) is always 1. In other words, consecutive terms in the
sequence are relatively prime (Grimaldi, 2012). Another pattern is shown by the addition of any
six consecutive terms; the sum of any six consecutive terms of the Fibonacci sequence is equal to
four times the value of the fifth term. Likewise, the sum of any eleven consecutive terms is always
divisible by 11 (Grimaldi, 2012). These are just a handful of the many interesting patterns that can
be found while investigating the Fibonacci sequence.
In this paper, more patterns will be explored, including patterns of generalized Fibonacci
sequences and Fibonacci sequences of higher order. Diagrams of the sequences will be studied,
and simple mathematical operations will be performed on different sequences. The produced
sequences will then be analyzed for patterns.
Fibonacci Sequences of Higher Orders 4
Literature Review
The Original Fibonacci Sequence
There are several ways to define a Fibonacci sequence besides the sum of the two previous
terms. One way is to describe any term in the list of numbers as equal to the term two steps ahead,
minus the term one step ahead (Fn=Fn+2-Fn-1). This is simply rearranging the original recursive
function. While similar, the new notation aids in proving that the sum of any terms 0 through n is
equal
to
one
less
than
the
term
that
corresponds
to
Fn+2
(for
example,
F0+F1+F2+F3+F4+F5=0+1+1+2+3+5=12 also equals F7-1 or 13-1) (Grimaldi, 2012). Another
formula used to describe the Fibonacci sequence that was popularized after Bigollo’s research is
(Fn+1)(Fn-1)=Fn2 + (-1)n ("Number game," 2013). Another way to write a formula to find the nth
term of the sequence involves the manipulation of the two solutions to the equation x2-x-1=0,
which are a=1/2 (1+√5) and b=1/2 (1-√5). These solutions have properties such that a+b=1, a-b=√5,
and ab=-1. These two solutions are related to the Fibonacci sequence golden ratio. These solutions
can be used to calculate the nth term of a Fibonacci sequence as follows: Fn=(an-bn)/√5 (Horadam,
1961). Many formulas are known to produce the Fibonacci sequence, each having their own
benefits for finding certain patterns.
Interest in the converging ratio between terms of the Fibonacci sequence led to one of the
most useful and well-known ratios in mathematics. The Fibonacci sequence converges to an
irrational number sometimes referred to as the golden number, or the golden ratio. This number,
½ (1+√5), is unique for several reasons. It is the positive root of the function y=x2-x-1, and it is
the only positive figure that has a reciprocal equal to 1 less than itself. The ratio can also be
Fibonacci Sequences of Higher Orders 5
described as the proportion that is equal to both the ratio between the larger part to the smaller part
of a divided segment and the ratio between the larger part to the whole segment. This perfect
symmetry makes it appealing to sight, and therefore the ratio is commonly found in art and
architecture. When the ratio, also known as phi, is raised to integer powers, it creates a new
sequence. This sequence has coefficients of the Fibonacci sequence (1, 1, 2, 3, 5, 8. . . ) as well as
constants of the Lucas sequence (1, 3, 4, 7, 11, 18, (xn=xn-1+xn-2)) ("Number game," 2013). This
sequence is a general Fibonacci sequence that has several applications in number theory.
Growing study in the ratio phi led investigators to find its presence in several places in
nature and art. Terms of the Fibonacci sequence are extremely prevalent in plant leaf and seed
numbers. Some of the lower value terms, such as 3, 5, and 8 are very common in the number of
leaves in a many different species of flowers. The spiraling rows of pine cones also correspond
with several higher numbers in the famous sequence (Moore, 2008). Rectangles drawn with this
ratio are also very pleasing to the eye. These rectangles are called golden rectangles, named after
the golden ratio. Inscribing squares in this rectangle continually creates more and more golden
rectangles. Connecting the corners of the squares forms a spiral similar to those found in many
environments and living organisms.
Fibonacci Sequences of Higher Orders 6
Figure 1. A logarithmic spiral that is commonly found in
shells. It can be formed by cubing a rectangle ("Number
games," 2013).
The ratio is also prevalent in temples, paintings, and human proportions ("Number game," 2013).
The Fibonacci sequence and phi are both well-studied topics due to their distinct presence in
nature.
The list of numbers popularized by Bigollo is not limited to the example of pairs of rabbits.
A Fibonacci sequence only requires that each new term is generated by the summation of the
previous two. Therefore, a Fibonacci sequence can include non-integers (0.1, 0.1, 0.2, 0.3, 0.5, 0.8,
1.3, 2.1. . .) and different start values (205, 205, 410, 615, 1025, 1640, 2665, 4305. . .).
Theoretically, a Fibonacci sequence can be any list of numbers in which the previous two terms
are added to generate the subsequent one (Moore, 2008).
Generalizations of the Fibonacci Sequence
Generalized Fibonacci sequences are varied sequences similar to the Fibonacci sequence
in which start terms are changed altogether, or multiplied by a constant. However, the recursive
function still must hold true that Fn+1=Fn + Fn-1 .The produced sequence has similar characteristics
to that of the Fibonacci, but also has its own patterns that can be generalized into certain equations
based on how the sequence was manipulated.
One way to vary the terms in the Fibonacci sequence is to designate two coefficients, p and
q, to the start terms. In a generalized Fibonacci sequence where H1=p and H2=p+q (and the
sequence produced appears as follows: p, p+q, 2p+q, 3p+2q, 5p+3q, 8p+5q) it is observed that
Hn= pFn+qFn-1, where Fn and Fn-1 are terms in the Fibonacci sequence. From this, two summation
Fibonacci Sequences of Higher Orders 7
formulas can be derived for the sums of even or the sum of odd terms in the H generalized
Fibonacci sequence:
n
∑ H2i = H2n+1 -p
represents the sum of the even terms in the p, p+q, 2p+q, 3p+2q,
5p+3q, 8p+5q sequence
i=1
n
∑ H2i = H2n - q
represents the sum of the odd terms in the p, p+q, 2p+q, 3p+2q,
5p+3q, 8p+5q sequence
i=1
Figure 2. Summation of even and odd terms in a p and q generalized Fibonacci sequence. (Adapted from Basin, 1963).
Another mathematician used a similar strategy to find the equation to find the nth
term of a general Fibonacci sequence. The initial values of the traditional Fibonacci sequence can
be changed to produce a similar sequence so that Hn=Hn-1+Hn-2 where H1=p and H2=q. p and q can
be any random number, and the sequence produced is p, p+q, 2p+q, 3p+2q, 5p+3q, 8p+5q… From
this, another equation can be produced to find term n of any general sequence: Hn=(1/2√5)(lan-mbn),
where l=2(p-qb), and m=2(p-qa). One special case of this is when p=qn where n is an integer such
that the sequence p{1, 1+n, 2+n, 3+n, 5+3n, 8+5n . . . } is produced (Horadam, 1961).
Generalizing sequences into k-generalized Fibonacci sequences yields sequences with
different converging ratios related to the integer k. k-Fibonacci numbers can be represented by the
recursion equation Fk,n+1=k Fk,n + Fk,,n-1. The starting values assumed for this equation are Fk,0 =
0 and Fk,1 =1. The converging ratios are the positive solutions to the equation r2 − kr − 1 = 0,
which is (k±√(k2+4))/2. If k=1, then the traditional Fibonacci sequence is produced and the ratio is
Fibonacci Sequences of Higher Orders 8
(1+√5)/2, also known as the Golden Ratio or Φ. If k=2, then the sequence converges to the Silver
Ratio, 1+√2, and if k=3 then the Bronze Ratio is produced, (3+√13)/2. The two solutions of the
equation mentioned above show certain properties: they multiply to yield a product of 1, add to
yield a sum of k, subtract to yield a difference of √(k2+4), and square to a product of k* σ + 1 (σ is
the positive solution to the equation) (Sergio, 2012).
Fibonacci Sequences of Higher Order
Similar to how Fibonacci sequences can have different start terms, they can also be raised
to higher powers by summing different numbers of initial terms. Given that the variable k is a
whole number greater than or equal to two, the value of k is used to denote the order of the
Fibonacci sequence produced, or the number of terms summed to generate the next term. If k=2,
the classical Fibonacci sequence is produced, if k=3 the Tribonacci sequence is produced, if k=4
the Tetranacci sequence is produced etc. The general trend to describe the sequences is the
following: Fn=Fn-1+Fn-2...+Fn-k, where there are k-1 zeroes preceding the first term, which equals
1. One important trend of this generalized sequence is that the first k+1 terms that do not equal
zero are increasing powers of two (Fkk+1=2k-1, Fkk+2=2k-1).
Table 1. From left to right, the order, name, and first few terms of each sequence (Bravo, 2013).
Fibonacci Sequences of Higher Orders 9
A characteristic polynomial equation can be assigned to the k-generalized Fibonacci
sequence (Fkn) where y(x)=xk-xk-1-. . . -x -1 (Bravo, 2013).
Discrete Dynamical Systems
A sequence defined by p terms given explicitly and a difference equation (recurrence
relation) for each term after the original terms is called a discrete dynamical system. The smallest
number of initial terms given with which the recurrence relation and the first p terms still hold is
called the order of the system ("Discrete dynamical systems," n.d.). For example, the original
Fibonacci sequence is of order 2 because it requires 2 initial start values.
A function that satisfies the explicit p terms and the recurrence relation collectively is
known as a solution to the discrete dynamical system. A function that contains p parameters and
still satisfies the recurrence relation is called a general solution to the discrete dynamical system.
The p parameters of the function can be varied to accommodate different initial values in different
discrete dynamical systems, and therefore it is a called a general solution ("Discrete dynamical
systems," n.d.).
Recurrence relations can be linear or non-linear. They are linear if there are p constants
and the recurrence relation can be written as such:
𝑎𝑛 = 𝑐1 𝑎𝑛−1 + 𝑐2 𝑎𝑛−2 + . . . + 𝑐𝑝 𝑎𝑛−𝑝 + 𝑓(𝑛)
If f(n) is a zero function, then the linear function is homogenous. If not, then the linear
function is non-homogenous. However, if the f(n) function is a constant value L for all values of
n, then the linear non-homogenous discrete dynamical system is called affine("Discrete dynamical
systems," n.d.).
Discrete dynamical systems of order 2 can be solved assuming that the solution is in the
form of crn. They key to solving the system is finding the polynomial characteristic equation of
Fibonacci Sequences of Higher Orders 10
that system. This equation can be derived from the recurrence relation, and once solved, can be
used in accordance with the initial values of the discrete dynamical system to find the solution for
the system. The Fibonacci sequence of order 2, given by the initial terms F0=1 F1=1 and the
recurrence relation Fn=Fn-1 + Fn-2, has the explicit solution
1 + √5
𝑛
1 + √5
−1 + √5 1 − √5
𝐹(𝑛) = (
)(
) +(
)(
)
2
2
2√5
2√5
𝑛
Although not it is not easy to see where this equation comes from directly, the process of
solving this discrete dynamical system is shown below:
𝐹1 = 1
𝐹0 = 1
𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2
Assume:
𝐹𝑛 = 𝑐𝑟 𝑛
Therefore:
𝑐𝑟 𝑛 = 𝑐𝑟 𝑛−1 + 𝑐𝑟 𝑛−2
𝑐𝑟 𝑛−2 (𝑟 2 − 𝑟 − 1) = 0,
There are two possible solutions that satisfy this equations:
𝑟=(
1 + √5
),
2
𝑠=(
Therefore:
𝑓(𝑛) = 𝐶1 𝑟 𝑛 + 𝐶0 𝑠 𝑛
1 = 𝐶1 𝑟 0 + 𝐶0 𝑟 0
1 = 𝐶1 𝑟1 + 𝐶0 𝑠1
1√5
)
2
Fibonacci Sequences of Higher Orders 11
𝐶1 =
1 + √5
2√5
,
1 + √5
𝐶0 =
−1 + √5
2√5
𝑛
1 + √5
−1 + √5 1 − √5
𝑓(𝑛) = (
)(
) +(
)(
)
2
2
2√5
2√5
𝑛
Figure 3. Mathematical solution adapted from "Discrete dynamical systems."
Mathematical Proofs
Mathematical proofs are the foundation of mathematical knowledge (J. Barys, personal
communication, February 9, 2014). In a proof, there are two main parts. The first part is a
hypothesis, which includes any assumptions or givens that are free to be manipulated. This part
often includes words like "let, suppose, assume, and if." The conclusion is the actual end goal of
what is being proven. In this process, via other theorems, proofs, or mathematical definitions, the
hypothesis must carry logically onto the conclusion. It is important to include everything in a steplike manner, so that the reader can follow the proof as the writer intended it to be interpreted
(variables should always be defined). Proofs have correct grammar, and therefore
equations/symbols should not start any sentences ("Ten tips for writing mathematical proofs," n.
d.).
There are two main implications that can be made from a mathematical proof. One is an ifthen statement, where the "if" states the hypothesis and the "then" presents the conclusion
(hypothesis ⇒ conclusion). There also is an “if and only if” statement (often written as iff), where
the hypothesis yields a conclusion, and similarly the conclusion yields the hypothesis (hypothesis
⇔ conclusion). Because of this, if one of the two is either true or false, the other must be the same.
Proof by contradiction is a strategy which uses the contrapositive to prove something true. Given
the example (a ⇒b)⇔(¬b⇒¬a), if one can prove that (¬b⇒¬a), then one has subsequently proven
Fibonacci Sequences of Higher Orders 12
that (a ⇒b). This is because the contrapositive of a true statement is always true (Ganesh, 2006).
Proof by induction is a mathematical process to prove a conjecture for all natural numbers
greater than or equal to a base number b. This involves proving the conjecture definitively for case
b. Then, if the case is true for some value n ≥ b, and it is also true for the case n+1, then the
conjecture is proven true. In words, a base step must be proven, then the inductive step must be
proven to show that case n allows the truth of case n+1. To prove P(n+1), it is often helpful to
relate that term to P(n), in order to utilize the assumption originally made (Cunningham, 2013).
Fibonacci Sequences of Higher Orders 13
Mathematical Objective
Mathematical Conjecture:
The converging ratio of consecutive terms of Fibonacci sequences of order k approaches 2
as k increases towards infinity. Related Fibonacci sequences of the same order k can be summed
and differenced to produce a sequence of the same order k where the initial terms are the sum or
difference of the initial terms used to generate the new sequence. Also, tree diagrams can be drawn
for Fibonacci sequences of any order k. These diagrams can be transcribed into numbers to show
that each term of order of the Fibonacci sequence of order k is a summation of different terms of
different sequences of that same order.
Mathematical Goal:
The goal of this project is to prove that the limit of the converging ratio of a k order
Fibonacci sequence is 2. The goal is also to show that sums of Fibonacci sequences of the same
order produce another Fibonacci sequence of that order by using the solutions to the discrete
dynamical systems of the different sequences. Proof by mathematical induction will be used to
prove a base case and the induction step to prove the conjecture for all orders of k. This proof will
then be related to the diagrams of Fibonacci sequences to see how the formation of all Fibonacci
tree diagrams can be made.
Fibonacci Sequences of Higher Orders 14
Methodology
Definitions:
Discrete Dynamical System (DDS): a DDS is a sequence of numbers (ak+1, ak+2, ak+3 … ak+j )
defined by initial terms (I) and a recurrence relation (R).
Initial Terms: The initial terms of a DDS are the terms given explicitly that start the sequence
of numbers.
Recurrence Relation: A recurrence relation relates previous terms of the DDS to the next term
in the sequence through one or more functions.
Solution to a Discrete Dynamical System: A solution to a DDS is a closed form equation that
explicitly defines all the terms in the sequence.
Order of a DDS: The order of a discrete dynamical system is the smallest number of initial
values needed for which the initial terms and the recurrence relation still hold true.
Geometric Sequence: A geometric sequence is a sequence of numbers in which the previous
term in the sequence is multiplied by a constant rate to yield the next. The rate remains the
same, and the sequence can be written as a DDS, and therefore there is a solution to satisfy that
discrete dynamical system.
Sum of a Geometric Sequence: The sum of the first n terms of a geometric sequence, from a1
to an, is expressed as:
𝑎1 (𝑟 𝑛 − 1)
𝑟−1
Fibonacci Sequences of Higher Orders 15
Fibonacci Sequence: The Fibonacci sequence is a DDS with initial terms and recurrence
relation shown below:
𝐹𝑛 = 𝐹𝑛−1 +𝐹𝑛−2 ,
𝐹0 = 1,
𝐹1 = 1
Generalized Fibonacci Sequence: A generalized Fibonacci sequence is a DDS with the same
recurrence relation as the Fibonacci sequence but with different initial values. These general
initial terms will be referred to as p and q or a and b.
Fibonacci Sequence of Higher Order: A Fibonacci sequence of higher order is a DDS similar
to that of the Fibonacci sequence. For this paper, the initial terms of these discrete dynamical
systems will be 1. However, the recurrence relation of the DDS sums the previous k terms of
the sequence to yield the next. This number k is called the order of the sequence.
Fibonacci Tree Diagram: A Fibonacci tree diagram is a visual representation of the Fibonacci
sequence (of order 2) in the form of a tree. There is a set rule for how the branches grow: each
branch grows two stages before producing additional branches besides itself. The maximum
number of branches each branch can grow is 2 (itself and one additional branch). At each stage
of growth, the sum of branches at that stage corresponds to the terms in the Fibonacci sequence.
In this paper, tree diagrams of Fibonacci sequences of higher orders will be investigated, and
thus new rules will be explained.
1.)
In this paper, previous math concepts were used to algebraically show patterns and
rules of different Fibonacci sequences. The definition of the sum of a geometric sequence was
used to in coordination with recurrence relations of DDSs to investigate the trend of converging
ratios of Fibonacci sequences of higher order. Algebraic manipulation and reasoning was
combined to show the sum of a geometric sequence with a constant rate of 2. This was then
Fibonacci Sequences of Higher Orders 16
related back to the DDSs of Fibonacci sequences to support the conjecture that the limit of the
converging ratio of Fibonacci sequences of increasing orders is 2.
2.)
The solution to the DDS of the Fibonacci sequence of order 2 was used along with
general start terms to prove that Fibonacci sequences of order 2 can be summed. First, a
solution to a Fibonacci DDS with general start terms was derived. This formula was added to
another Fibonacci DDS with general start terms to show that the result is a new Fibonacci
sequence of order 2. Finally, this concept was reinforced with the recurrence relations of the
discrete dynamical systems, and expanded to Fibonacci sequences of higher orders.
3.)
Fibonacci tree diagrams of higher orders were drawn and the rules to draw the trees
were analyzed. The diagrams were then be transcribed into tables to see the patterns more
directly. Then, induction was used to aid in explaining why Fibonacci tree diagrams of higher
orders can be produced.
Fibonacci Sequences of Higher Orders 17
Results
1.) Limit of Converging Ratios: the limit of the converging ratio of a Fibonacci sequence is 2.
Given the Fibonacci sequence of order k:
𝑎𝑛 = 𝑎𝑛−1 + 𝑎𝑛−2 +. . . +𝑎𝑛−𝑘
Assuming that the system is exponential, the conjecture can be made that 𝑎𝑛 = 𝑐𝑟 𝑛 in which c is
a constant and r is a rate. Thus, the substitution is made:
𝑐𝑟 𝑛 = 𝑐𝑟 𝑛−1 + 𝑐𝑟 𝑛−2 + 𝑐𝑟 𝑛−3 +. . . +𝑐𝑟 𝑛−𝑘
𝑐𝑟 𝑛 − 𝑐𝑟 𝑛−1 − 𝑐𝑟 𝑛−2 − 𝑐𝑟 𝑛−3 −. . . −𝑐𝑟 𝑛−𝑘 = 0
𝑐𝑟 𝑛−𝑘 (𝑟 𝑘 − 𝑟 𝑘−1 − 𝑟 𝑘−2 −. . . −𝑟 − 1) = 0
In order for this to be true, there are k possible values for r:
𝑟1 = 𝑎,
𝑟2 = 𝑏,
𝑟3 = 𝑐,
…
𝑟𝑘 = 𝛾
The rate with the greatest magnitude will be the converging ratio of that Fibonacci
sequence. Because all of the sequences that are dealt with in this paper are positive, this converging
ratio will also be positive. A graph can be made to show the real solutions of the characteristic
equation used to find the rates:
Fibonacci Sequences of Higher Orders 18
Figure 3. A graph of the characteristic equations of Fibonacci sequences from order 2 to order 10.
As shown, the positive roots of the equations (which also are the rates of the greatest
magnitudes) grow closer and closer to 2, but they will never reach that value. It appears that all of
the graphs intersect (and will always intersect) at the coordinate (2, 1). Thus, the conjecture was
made for all positive integers n that
2𝑛 − 2𝑛−1 − 2𝑛−2 −. . . −20 = 1,
which can be rewritten as
2𝑛 = 1 + 20 + 21 + 23 +. . . +2𝑛−1 .
This conjecture can be shown true both with the definition of the sum of a geometric sequence and
by proof by mathematical induction. The equation for the sum of any geometric sequence is
𝑛
∑ 𝑐𝑟
𝑖−1
𝑖=1
𝑐𝑟 0 (𝑟 𝑛 − 1)
=
𝑟−1
The rates and constants can be substituted such that
𝑛
𝑖−1
∑ 1(2)
𝑖=1
(1)20 (2𝑛 − 1)
=
2−1
𝑛
∑ 1(2)𝑖−1 =
𝑖=1
1(2𝑛 − 1)
1
𝑛
∑ 1(2)𝑖−1 = (2𝑛 − 1)
𝑖=1
If the sigma notation is replaced with the entire sum of the products, it is shown that:
20 + 21 +. . . +2𝑛−1 = (2𝑛 − 1)
When 1 is added to both sides, the conjecture is shown as true:
1 + 20 + 21 +. . . +2𝑛−1 = 2𝑛
Similarly, the conjecture can be proven with proof by mathematical induction:
Fibonacci Sequences of Higher Orders 19
Conjecture:
1 + 20 + 21 +. . . +2𝑛−1 = 2𝑛
For case n=1:
1 + 21−1 = 21
1+1 =2
2=2
Assume true for any positive integer n = k that:
1 + 20 + 21 +. . . +2𝑘−1 = 2𝑘
Prove for case n = k + 1 that:
1 + 20 + 21 +. . . +2𝑘−1 + 2𝑘 = 2𝑘+1
The assumption made can be substituted for the expression “1 + 20 + 21 +. . . +2𝑘−1 ” such that
2𝑘 + 2𝑘 = 2𝑘+1
2𝑘 (1 + 1) = 2𝑘+1
2𝑘 (2) = 2𝑘+1
2𝑘+1 = 2𝑘+1
It can now be assumed that the graphs will intersect at the coordinate (2, 1). Because of
this, the functions will have positive roots that approach 2. Consequently, the converging ratios of
Fibonacci sequences as the order of the sequences increase will approach 2.
2.) Sums of Fibonacci Sequences: Fibonacci sequences of the same order sum to another sequence
of that same order.
Investigation in Fibonacci tree diagrams led to the conjecture that Fibonacci sequences of
the same order can be summed to produce another Fibonacci sequence of that order in which the
Fibonacci Sequences of Higher Orders 20
start terms are the sums of the respective start terms used to produce that sequence. This can be
proven by finding the explicit function of a Fibonacci sequence of order 2 with arbitrary start
terms. Given the following recursive relation and initial values of a discrete dynamical system:
𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2
𝐹0 = 𝑎
𝐹1 = 𝑏
The conjecture can be made that solution to this discrete dynamical system is exponential, because
the ratio consecutive terms converges to one value. So, the solution could be in the form of 𝐹𝑛 =
𝑐𝑟 𝑛 . Substitution is conducted such that:
𝑐𝑟 𝑛 = 𝑐𝑟 𝑛−1 + 𝑐𝑟 𝑛−2
𝑐𝑟 𝑛 − 𝑐𝑟 𝑛−1 − 𝑐𝑟 𝑛−2 = 0
𝑐𝑟 𝑛−2 (𝑟 2 − 𝑟 − 1) = 0
There are two possible solutions of r that could make this equation true:
𝑟1 =
1 + √5
,
2
𝑟2 =
1 − √5
2
Now, a conjecture could be made that the explicit function is in the form:
𝐹(𝑛) = 𝑐1 𝑟1 + 𝑐2 𝑟2
The constants can be found using the initial terms of the system:
𝐹(𝑛) = 𝑐1 𝑟1 + 𝑐2 𝑟2
𝐹(𝑛) = 𝑐1 𝑟1 + 𝑐2 𝑟2
𝐹(0) = 𝑐1 𝑟10 + 𝑐2 𝑟20
𝐹(1) = 𝑐1 𝑟11 + 𝑐2 𝑟21
𝑎 = 𝑐1 + 𝑐2
b = c1 (
With this system of equations, the constants are found:
1 + √5
1 − √5
) + c2 (
)
2
2
Fibonacci Sequences of Higher Orders 21
𝑐1 =
𝑎(−1 + √5) + 2𝑏
2√5
, 𝑐2 =
𝑎(1 + √5) − 2𝑏
2√5
and thus the explicit function is found:
𝐹 (𝑛 ) = (
𝑎(−1 + √5) + 2𝑏
2√5
𝑛
1 + √5
𝑎(1 + √5) − 2𝑏 1 − √5
)(
) +(
)(
)
2
2
2√5
𝑛
The same process can be used to find the explicit function for a similar discrete dynamical system
with initial values p and q:
𝐺𝑛 = 𝐺𝑛−1 + 𝐺𝑛−2
𝐺0 = 𝑝
𝐺1 = 𝑞
𝐺 (𝑛 ) = (
𝑝(−1 + √5) + 2𝑞
2√5
𝑛
1 + √5
𝑝(1 + √5) − 2𝑞 1 − √5
)(
) +(
)(
)
2
2
2√5
𝑛
Now, a new function is created to sum the two explicit functions that were just derived:
𝐻 (𝑛) = 𝐹 (𝑛) + 𝐺(𝑛)
𝐻 (𝑛 ) = (
𝑎(−1 + √5) + 2𝑏
2√5
+(
𝑛
1 + √5
𝑎(1 + √5) − 2𝑏 1 − √5
)(
) +(
)(
)
2
2
2√5
𝑝(−1 + √5) + 2𝑞
2√5
𝑛
1 + √5
𝑝(1 + √5) − 2𝑞 1 − √5
)(
) +(
)(
)
2
2
2√5
𝑛
1 + √5
𝑎(−1 + √5) + 2𝑏 𝑝(−1 + √5) + 2𝑞
𝐻 (𝑛 ) = (
) (
+
)
2
2√5
2√5
𝑛
𝑛
1 − √5
𝑎(1 + √5) − 2𝑏 𝑝(1 + √5) − 2𝑞
+(
) (
+
)
2
2√5
2√5
𝑛
Fibonacci Sequences of Higher Orders 22
𝑛
1 + √5
(𝑎 + 𝑝)(−1 + √5) + 2(𝑏 + 𝑞)
𝐻 (𝑛 ) = (
) (
)
2
2√5
𝑛
1 − √5
(𝑎 + 𝑝)(1 + √5) − 2(𝑏 + 𝑞)
+(
) (
)
2
2√5
After factoring, the simplified function is in the same format as the two functions added to
yield it. It has the same rates, and the constants associated with those rates are similar to the ones
used to produce the sequence. However, where the initial terms should be, there now is a sum of
the initial terms used to produce the sequence. It thus can be deduced that this new function, the
sum of two Fibonacci sequences of order two, is also a Fibonacci sequence, and the initial terms
are equal to the sums of the initial terms of the two sequences added. This can be extended to
Fibonacci sequences of higher orders:
Given the Fibonacci sequence of order k:
𝑎𝑛 = 𝑎𝑛−1 + 𝑎𝑛−2 +. . . +𝑎𝑛−𝑘
Because the sequence is exponential, the conjecture can be made that:
𝑎𝑛 = 𝑐𝑟 𝑛
in which c is a constant and r is a rate. Thus, the substitution is made:
𝑐𝑟 𝑛 = 𝑐𝑟 𝑛−1 + 𝑐𝑟 𝑛−2 + 𝑐𝑟 𝑛−3 +. . . +𝑐𝑟 𝑛−𝑘
𝑐𝑟 𝑛 − 𝑐𝑟 𝑛−1 − 𝑐𝑟 𝑛−2 − 𝑐𝑟 𝑛−3 −. . . −𝑐𝑟 𝑛−𝑘 = 0
𝑐𝑟 𝑛−𝑘 (𝑟 𝑘 − 𝑟 𝑘−1 − 𝑟 𝑘−2 −. . . −𝑟 − 1) = 0
In order for this to be true, there are k possible values for r:
𝑟1 , 𝑟2 , 𝑟3 … 𝑟𝑘
Thus, the explicit function is
𝑎(𝑛) = 𝑐1 𝑟1 + 𝑐2 𝑟2 + 𝑐3 𝑟3 +. . . +𝑐𝑘 𝑟𝑘
Fibonacci Sequences of Higher Orders 23
where the constants are derived from the k given initial terms. Another explicit function of a
Fibonacci sequence of that same order can be found in a similar way, where the constants are
derived from different start values. Thus, this other function would be
𝑏(𝑛) = 𝐶1 𝑟1 + 𝐶2 𝑟2 + 𝐶3 𝑟3 +. . . +𝐶𝑘 𝑟𝑘 .
A function of the sum of these two explicit functions can written:
𝑐 (𝑛) = 𝑎(𝑛) + 𝑏(𝑛)
𝑐 (𝑛) = 𝑐1 𝑟1 + 𝑐2 𝑟2 + 𝑐3 𝑟3 +. . . +𝑐𝑘 𝑟𝑘 + 𝐶1 𝑟1 + 𝐶2 𝑟2 + 𝐶3 𝑟3 +. . . +𝐶𝑘 𝑟𝑘
𝑐 (𝑛) = (𝑐1 +𝐶1 )𝑟1 + (𝑐2 + 𝐶2 )𝑟2 + (𝑐3 + 𝐶3 )𝑟3 +. . . +(𝑐𝑘 + 𝐶𝑘 )𝑟𝑘
Because the constants are derived from the same rates, they can be simplified to a new constant
that will yield a new Fibonacci sequence of that order.
This conclusion can more clearly be seen by the addition of the original recursive formulas:
𝑐 (𝑛 ) = 𝑎 (𝑛 ) + 𝑏 (𝑛 )
𝑐𝑛 = 𝑎𝑛−1 + 𝑎𝑛−2 +. . . +𝑎𝑛−𝑘 + 𝑏𝑛−1 + 𝑏𝑛−2 +. . . +𝑏𝑛−𝑘
Because each c term is the addition of two corresponding a and b terms, each a and b of the same
position can be simplified to a c term of that position:
𝑐𝑛 = 𝑎𝑛−1 + 𝑏𝑛−1 + 𝑎𝑛−2 + 𝑏𝑛−2 +. . . +𝑎𝑛−𝑘 + 𝑏𝑛−𝑘
𝑐𝑛 = 𝑐𝑛−1 + 𝑐𝑛−2 +. . . +𝑐𝑛−𝑘
This recursion relation describes a new Fibonacci sequence of the same order k.
Because Fibonacci sequences are a series of added numbers, the addition of Fibonacci
sequences is additive and commutative. For example:
𝑐 (𝑛 ) = 𝑏 (𝑛 ) + 𝑎 (𝑛 )
𝑐𝑛 = 𝑏𝑛−1 + 𝑏𝑛−2 +. . . +𝑏𝑛−𝑘 + 𝑎𝑛−1 + 𝑎𝑛−2 +. . . +𝑎𝑛−𝑘
𝑐𝑛 = 𝑏𝑛−1 + 𝑎𝑛−1 + 𝑏𝑛−2 + 𝑎𝑛−2 +. . . +𝑏𝑛−𝑘 + 𝑎𝑛−𝑘
Fibonacci Sequences of Higher Orders 24
𝑐𝑛 = 𝑐𝑛−1 + 𝑐𝑛−2 +. . . +𝑐𝑛−𝑘
Each term in the sequence is added, so the terms can be rearranged in any order to produce the
same exact Fibonacci sequence. Similarly, if three sequences are being added, the three terms of
the same position define the single term in the new sequence produced. Because of that, the same
process can be done to simplify the recursion relation such that the new sequence is also a
Fibonacci sequence. This can be done for any number of Fibonacci sequences in any order.
3.) Fibonacci Tree Diagrams: Fibonacci tree diagrams can be drawn for any Fibonacci sequence
order.
For any Fibonacci sequence of order k, there exists a tree diagram to represent it. This
diagram has k+1 different types of branches. These types will be described as either matureness
level or matureness stage, a ranking that ranges from 1 to k+1. The matureness level of the branch
increases by one each time the height of the tree increases by one. The first two matureness levels
of the tree will always be dormant; they create no additional branches.
Each subsequent
matureness level produces one more branch (besides itself) than the previous matureness level.
These new branches are new, and begin at matureness level one. At the maximum matureness
level, k+1, the branch can no longer mature. It stays at this level forever. In this paper, all trees
begin with 1 initial branch in the first maturity level. Refer to figures 3-5 for the Fibonacci tree
diagrams of Fibonacci sequences of orders 2-4.
The diagrams of the sequences can easily be converted into tables of the number of
branches at each matureness level. This table allows for ease of visibility of trends (refer to tables
Fibonacci Sequences of Higher Orders 25
2-5). The tables also allow for creating a tree diagram without actually drawing the tree. Below is
the tree diagram table of the Fibonacci sequence of order 8:
Table 2. The Fibonacci tree diagram table of a Fibonacci sequence of order 8.
Matureness
Height
1
2
3
4
5
6
7
8
9
1
1
0
0
0
0
0
0
0
0
2
0
1
0
0
0
0
0
0
0
3
1
0
1
0
0
0
0
0
0
4
2
1
0
1
0
0
0
0
0
5
4
2
1
0
1
0
0
0
0
6
8
4
2
1
0
1
0
0
0
7
16
8
4
2
1
0
1
0
0
8
32
16
8
4
2
1
0
1
0
9
64
32
16
8
4
2
1
0
1
10
127
64
32
16
8
4
2
1
1
11
254
127
64
32
16
8
4
2
2
12
507
254
127
64
32
16
8
4
4
13
1012
507
254
127
64
32
16
8
8
Σ
1
1
2
4
8
16
32
64
128
255
509
1016
2028
There are a few trends to see in this table. Each row is a copy of the previous row, just
shifted one place to the right. This is because each branch grows to the next matureness level at
every increment of height. Also, the number of the branches at the first maturity level is the sum
of the number of branches at the third, two times the branches at the fourth, three times the branches
at the fifth, etc. These rules will apply later in the paper. The following three pages show the
Fibonacci tree diagrams and tables of orders 2, 3, and 4.
Fibonacci Sequences of Higher Orders 26
Figure 3. A Fibonacci tree diagram that represents a Fibonacci sequence of order two. The different colors represent different levels of matureness in each branch. The
dashed blue lines mark the total height, with the first line representing a height of 0.
height of the table extends past that of the diagram to better show trends.
Table 3. A Fibonacci tree diagram transcribed into a table of values. The matureness of a branch is represented by the row types (blue, pink, red), and the
number of that type of branch is recorded over different heights. The total number of branches sums to the Fibonacci sequence of order 2. The range of the
height of the table extends past that of the diagram to portray certain trends.
Matureness
Height
1 (blue)
2 (pink)
3 (red)
1
1
0
0
2
0
1
0
3
1
0
1
4
1
1
1
5
2
1
2
6
3
2
3
7
5
3
5
8
8
5
8
9
13
8
13
10
21
13
21
11
34
21
34
12
55
34
55
13
89
55
89
Σ
1
1
2
3
5
8
13
21
34
55
89
144
233
Fibonacci Sequences of Higher Orders 27
Figure 4. A Fibonacci tree diagram of order 3. The color of branch represents matureness of that branch. This diagram has one more level of matureness than the diagram of the Fibonacci
tree of order 2. Similarly, the blue dashed lines mark the height of the tree, with the first line representing a height of 0.
Table 4. The Fibonacci tree diagram of order 3 transcribed into a table of values. Similar to the previous table, this table shows the number of branches at each matureness as well
as the sum of all the branches at each height. The range of the height of the table extends past that of the diagram to better show trends.
Matureness
Height
1 (blue)
2 (pink)
3 (red)
4 (green)
1
1
0
0
0
2
0
1
0
0
3
1
0
1
0
4
2
1
0
1
5
3
2
1
1
6
6
3
2
2
7
11
6
3
4
8
20
11
6
7
9
37
20
11
13
10
68
37
20
24
11
125
68
37
44
12
230
125
68
81
13
423
230
125
149
Σ
1
1
2
4
7
13
24
44
81
149
274
504
927
Fibonacci Sequences of Higher Orders 28
Figure 5. A Fibonacci tree of order 4. The color of the branch designates the matureness of that branch, and the dashed blue lines mark the height
of the tree, with the first blue dashed line representing 0.
Matureness
Table 5. A Fibonacci tree diagram of order 4 converted into a table of values. The values show the number of branches at each matureness and the total number of branches at each height. The range
of the height of the table extends past that of the diagram to better show trends.
Height
1 (blue)
2 (pink)
3 (red)
4 (green)
5 (orange)
1
1
0
0
0
0
2
0
1
0
0
0
3
1
0
1
0
0
4
2
1
0
1
0
5
4
2
1
0
1
6
7
4
2
1
1
7
14
7
4
2
2
8
27
14
7
4
4
9
52
27
14
7
8
10
100
52
27
14
15
11
193
100
52
27
29
12
372
193
100
52
56
13
717
372
193
100
108
Σ
1
1
2
4
8
15
29
56
108
208
401
773
1490
Fibonacci Sequences of Higher Orders 29
It appears that for any Fibonacci tree diagram of any order k, the number of branches at
each matureness stage produces a Fibonacci sequence of that same order. In order for this to be
true, then the number branches in the first maturity stage must produce a Fibonacci sequence of
that same order k as height increases. If the first stage satisfies this requirement, then the number
of branches in the rest of the maturity levels will also be Fibonacci sequences of that same order.
This is because each subsequent maturity level is produced by the number of branches of the
previous maturity level from the previous height. As seen in the tables above, to create the number
of branches in a maturity level other than the first, the number of branches from the previous
maturity level at the previous height (the cell to the left and up) is translated down and to the right.
Therefore, the maturity levels between the first and last are copies of the exact copies of the first.
The last maturity level is an exception to the rule. It is created by summing the cell to the left and
the cell to the left and up. This sum of two Fibonacci sequences will produce another, as previously
shown. So, if the first maturity level is a Fibonacci sequence, then the number of branches at each
maturity level is.
In order to show that the first maturity level is a Fibonacci sequence, it must be proven
that the first terms are 1, 0, 20, 21, 22… 2k-2, 2k-1 -1. Given that the first k+2 non-zero terms of a
Fibonacci sequence of order k are 1, 1, 2, 4… 2k-1, 2k-1 (Bravo, 2013), it was deduced that if the
first non-zero terms started as 1, 0, 1, 2, 4… then the first terms k+2 terms of the sequence would
be 1, 0, 1, 2, 4… 2k-2, 2k-1-1. This is because when a zero is placed after the first non-zero term
rather than before it, the recursion of that Fibonacci sequence will stray from the powers of two
one term quicker because it will reach that 0 quicker than the other sequence. For example, given
the Fibonacci sequences of order 4 below:
𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 + 𝐹𝑛−3 + 𝐹𝑛−4
Fibonacci Sequences of Higher Orders 30
𝐹0 = 0, 𝐹1 = 0, 𝐹2 = 0, 𝐹3 = 1
It can be seen that the first k+2 non-zero terms are 1, 1, 2, 4, 8, 15. However, if the initial terms
were
𝐹0 = 0, 𝐹1 = 0, 𝐹2 = 1, 𝐹3 = 0
The first k+2 non-zero terms would be 1, 0, 1, 2, 4, 7, the values given by the rule stated above.
This will be assumed to be true for all order of Fibonacci sequences.
The first step to showing that the number of branches of the first maturity level start as 1,
0, 1, 2, 4… 2k-2, 2k-1-1, is to show that given the first two inputs are 1, 0, then the next k-1 terms
will be increasing powers of 2. By observing the formation of the number of branches in the first
maturity level, a conjecture can be made and proven by mathematical induction. This value is
produced by adding the number of branches at maturity level 3 with twice the amount of the
branches at maturity level 4, three times the amount at maturity level 5, four times the amount of
the branches at maturity level 6, etc. For example, to get the number of branches at maturity level
1 at height 6 of the Fibonacci sequence of order 8 (refer to table 2), the following operation is
conducted:
1(4) + 2(2) + 3( 1) + 4(0) + 5(1) = 8
Thus, the conjecture was made and proven with induction:
Conjecture:
1(2𝑛 ) + 2(2𝑛−1 ) + 3(2𝑛−2 )+. . . +(𝑛 + 1)(20 ) + (𝑛 + 2)(0) + (𝑛 + 3)(1) = 2𝑛+2
(the first term being added refers to the number of branches being produced by maturity level 3,
the next is the amount being produced by maturity level 4, etc.):
Case n=1:
1(21 ) + (1 + 1)(20 ) + (1 + 2)(0) + (1 + 3)(1) = 21+2
Fibonacci Sequences of Higher Orders 31
2 + 2 + 0 + 4 = 23
8=8
Assume true for any positive integer n=k:
1(2𝑘 ) + 2(2𝑘−1 ) + 3(2𝑘−2 )+. . . +(𝑘 + 1)(20 ) + (𝑘 + 2)(0) + (𝑘 + 3)(1) = 2𝑘+2
Prove true for n=k+1 that:
1(2𝑘+1 ) + 2(2𝑘 ) + 3(2𝑘−1 )+. . . +(𝑘 + 2)(20 ) + (𝑘 + 3)(0) + (𝑘 + 4)(1) = 2𝑘+3
By substituting the assumption made for n=k, this equation is left:
(2𝑘+2 ) + (2𝑘+1 ) + (2𝑘 ) + (2𝑘−1 )+. . . +(20 ) + (1) = 2𝑘+3
This is known to be true from the induction proof from section one of the results section. So:
2𝑘+3 = 2𝑘+3
It has now been shown that given the first to values in the first maturity level of the Fibonacci tree
diagram as 1, 0, the rest will be increasing powers of 2, as expected.
The next step is to show that the next number in the sequence will be one less than a power
of two. By observing the formation of the Fibonacci tree table, it can deduced how this term is
created. For example, to get the value for the first maturity level at height 10, (refer to table 2), the
following calculation is performed:
7(1) + 6(1) + 5(2) + 4(4) + 3(8) + 2(16) + 1(32) = 127
To generalize the rule of forming the first number that is 1 less than a power of 2, the following
conjecture was made and proven by induction:
Conjecture:
𝑛(1) + (𝑛 − 1)(20 ) + (𝑛 − 2)(21 ) + (𝑛 − 3)(22 )+. . . +(1)(2𝑛−2 ) = 2𝑛 − 1
(n refers to the number of branches produced by the maximum maturity level. The following terms
are the products of the number of branches in decreasing maturity levels and the number of
Fibonacci Sequences of Higher Orders 32
additional branches they produce, and the sum will produce the number of branches in the first
maturity level):
Case n=2:
2(1) + (2 − 1)(22−2 ) = 22 − 1
2 + (1)(20 ) = 4 − 1
2+1 =4−1
3=3
Assume true for any positive integer greater than or equal to 2, n=k:
𝑘(1) + (𝑘 − 1)(20 ) + (𝑘 − 2)(21 ) + (𝑘 − 3)(22 )+. . . +(1)(2𝑘−2 ) = 2𝑘 − 1
Prove true for n=k+1:
(𝑘 + 1)(1) + (𝑘)(20 ) + (𝑘 − 1)(21 ) + (𝑘 − 2)(22 )+. . . +(1)(2𝑘−1 ) = 2𝑘+1 − 1
(𝑘 + 1)(1) + 𝑘(1) + 2(𝑘 − 1)(20 ) + 2(𝑘 − 2)(21 )+. . . +(2)(1)(2𝑘−2 ) = 2𝑘+1 − 1
(2𝑘 − 1) + (𝑘 + 1)(1) + (𝑘 − 1)(20 )+. . . +(1)(2𝑘−2 ) = 2𝑘+1 − 1
(2𝑘 − 1) + 1 + 𝑘 + (𝑘 − 1)(20 )+. . . +(1)(2𝑘−2 ) = 2𝑘+1 − 1
(2𝑘 − 1) + 1 + (2𝑘 − 1) = 2𝑘+1 − 1
2𝑘 + 2𝑘 − 1 = 2𝑘+1 − 1
2𝑘+1 − 1 = 2𝑘+1 − 1
It has now been shown that the first terms of the first maturity level will be 1, 0, 1, 2, 4…
2k-2, 2k-1-1. Consequently, the number of branches in the first maturity level and thus the number
of branches in all of the maturity levels produce Fibonacci sequence. Because the total number of
branches of the tree at any one height is the sum of multiple Fibonacci sequences, the total number
of branches also produces a Fibonacci sequence. Therefore, assuming that the rules designated
before are followed, a Fibonacci tree diagram can be made of any order.
Fibonacci Sequences of Higher Orders 33
Discussion
Conclusions
This study showed that the converging ratios of Fibonacci sequences of order k approaches
2 as k increases to infinity. With the characteristic equations of Fibonacci sequences of higher
order, induction was used to show that the converging ratio of consecutive terms has a limit of 2.
Also, it was shown that Fibonacci sequences of the same order can be summed to produce another
sequence of that same order. This was shown with the explicit functions of Fibonacci sequences
as well as the recursive relations of Fibonacci sequences. Consequently, it was shown that given
certain parameters, Fibonacci tree diagrams of any order can be created because the constituent
parts of the diagrams are sums of Fibonacci sequences. The rules to create a Fibonacci tree diagram
if a Fibonacci sequence of order k are as follows:
There are k+1 types of branches, described by a maturity level of 1 through k+1. Branches
mature directly from one maturity level to the next as the height of the tree increases by one. The
tree begins with one branch at a maturity level of 1. Maturity levels 1 and 2 are dormant; they
produce no additional branches. Each subsequent maturity level produces one additional branch
(besides itself) than the one before. Maturity level 3 produces 1 additional branch, maturity level
4 produces 2 additional branches, etc. When an additional branch is produced, it begins at maturity
level 1. When the maximum maturity level is reached and the branch can no longer mature; it stays
at this stage indefinitely as height continues.
Fibonacci Sequences of Higher Orders 34
Limitations and Assumptions
There were a few assumptions made in this study. First, it was assumed that Fibonacci
sequences continue on indefinitely according to a recursive relation. Another assumption made
was that Fibonacci sequences can be defined as exponential functions. This assumption was made
in other resources in order to find explicit functions of discrete dynamical systems, and thus this
process was mirrored in this paper. Also, there were two main parts of this paper that were deduced
from logic rather than out of mathematical proof. First, it was reasoned that the first k+1 terms of
a Fibonacci sequence that begins as 1, 0 will be 1, 0, 1, 2, 4… 2 k-2, 2k-1-1. This conjecture was
adapted from one theorem of a math paper used as a resource; however, this conjecture was not
proven due to time constraints. The other main assumption was that if the values in the first
maturity level of a Fibonacci tree diagram form a Fibonacci sequence, then the rest of the maturity
levels and thus the tree will form Fibonacci sequences. This conjecture was explained in detail in
the results, but it was not proven mathematically.
The main limitation of this study was the time constraint of the experiment. If there was
more time, some of the assumption made would have been proven. More conjectures would have
been made, and more patterns would have been found.
Fibonacci Sequences of Higher Orders 35
Applications and Future Experiments
This experiment further expands interesting patterns in Fibonacci-like sequences.
Fibonacci numbers and ratios are found virtually everywhere. However, Fibonacci sequences of
higher order are not as developed as the Fibonacci sequence of order 2. This study provides a
glimpse into the patterns that such sequences have. There are undoubtedly many more to be found,
but there is not enough research in the sequences to reveal the patterns. With more investigation
in the field, more trends could be found, and more connections to the natural environment could
be made.
In a future experiment, the patterns of the products of Fibonacci sequences could be
studied. The product of two Fibonacci sequences does not yield another sequence, but the new
sequence still has unique properties. These properties could be investigated and reported on in a
later investigation. This idea could also be expanded to quotients of Fibonacci sequences. Another
interesting investigation would be to study the patterns of sequences where previous terms are
multiplied instead of added. There could possibly be more ratios and interesting rules in sequences
defined as recursive products.
Fibonacci Sequences of Higher Orders 36
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Fibonacci Sequences of Higher Orders 38
Acknowledgements
The author would like to thank all of the faculty at Massachusetts Academy of Math and
Science for continued support throughout the experiment. Specifically, he would like to thank his
STEM advisor Mr. Thomas Regele for encouraging the author to pursue a project in mathematics,
and for continued support through the research phase. In addition, he would like to thank Mrs.
Maria Borowski for aiding in the organization of a math research paper. Furthermore, the author
wishes to thank Mr. Bill Ellis for the overseeing and organization of the STEM experience. Lastly,
the author would like to thank his parents for support through the research phase.