The Fibonacci sequence is a summation series created by adding the last two numbers of the sequence for the next number in the sequence.
This requires two starting numbers, or seed numbers, to begin the sequence. The classical Fibonacci sequence begins with a pair of ones (1 and 1).
A generalized Fibonacci sequence may begin with any pair of numbers, yet many of the qualities of the summation sequence endure.
The mathematical notation of the Fibonacci Sequence, where 'n' is the index to the sequence position:
F = 1, F = 1, F[n] = F[n-1] + F[n-2], … (continuing with increasing indices [n])
Son of a good guy
“In today's opinion, the work of Fibonacci marks the rebirth of Western mathematics. [...] It was the new beginning of the research on the history of mathematics in the early 19th Century in Italy, which led to a rediscovery of Fibonacci, and therein a recognition of his great importance in the history of mathematics.”
Leanardo of Pisa is the first person known to have mentioned the summation sequence in a mathematical problem concerning rabbit population. One rabbit plus one more rabbit (2) produces a third rabbit, etc. as 1, 1, 2, 3, 5, 8, 13, ...
As a son traveling with his father, who was a trade emissary for the Italian city of Pisa, Leonardo was exposed to Arabian decimal arithmetic.
Leonardo wrote of the rabbit population summation series in his book that brought Europe out of Roman Numeral mathematics, Liber Abaci (Book of Calculation), published in 1202. Almost seven centuries later, in the 1870s, the name 'Fibonacci' was applied to Leonardo by a mathematician Edouardo Lucas, who coined that name in books he wrote on the Fibonacci sequence. Leonardo's father was called Bonacci (meaning good natured man). Filius Bonacci is the Latin for son of Bonacci, shortened to Fibonacci.
Roughly translating, Fibonacci means a good man's son.
Summation from zero
Sometimes mathematicians begin the Fibonacci Sequence with zero.
The universal constant of the Golden Ratio, an approximation of Phi, or Φ, is found as the quotient between any adjacent two numbers (neighboring) in the Fibonacci sequence with increasing precision as the numeric sequence grows larger. The division of any adjacent pairs of the sequence approximate Phi, or (Phi - 1) when divided:
233 ÷ 144 = 1.6180555… =~ Phi (approx. within 0.00002)
Imploding Phi —a tabulation and graphs of a special seed pair of oppositely signed Phi degrees, and graphs of the sequence and the quotients of adjacent pairs.
This article Imploding Phi exemplifies one of many general Fibonacci sequence seed-pairs that produce a non-typical Fibonacci Sequence —Rather than grow larger exponentially, this 'generalized Fibonacci sequence' oscillates while approaching zero, then just prior to becoming zero, the identity at play is lost to decimal round-off near the computational limit of the spreadsheet, after which the summation sequence proceeds with typical exponential growth.
This very same behavior is found in the analytical equation that produces the general Fibonacci sequence along the real number line. See Binet's Formula below.
Moreover, the quotient of neighboring numbers in this atypical general Fibonacci sequence of numbers, when the summation sequence approaches zero, flips from -(Phi)-1 to Phi (-0.618 to 1.618). This occurs when alternate signs of certain powers of the golden ratio are used as seed numbers to start the sequence.
Expand the said impulse event into an X-Y mapping as a 2D surface using Binet's Forumla to generate a generate surface texture with a third dimension set as the iteration-depth of the impulse event —or some such threshold mapping, very similar to generating a Mandelbrot X-Y map colored by iterations of an orbit within selection-thresholds.
This thought is inspired by work by Prof. Reiter, about an escape basin in iterated golden ratio functions.
Table 1. The digital roots of the Fibonacci sequence.
Another modulo pattern is found in the Fibonacci sequence which is not technically a modulo (divide by N), as the zero element is not used. This is a pattern in the digital roots of the sequence. The root is found by adding the digits to obtain one remaining number as one digit.
Examples of digital root compression:
Compress the 13th Fibonacci number, 233, into a digital root:
Compress the 17th Fibonacci number, 1597, into a digital root:
1+5=6, 6+9=15 which is 1+5=6, and finally, 6+7=13, which is 1+3=4
Curiously, the digital roots of the second group of 24 Fibonacci numbers are identical to the first 24 digital roots of the Fibonacci sequence. The pattern of the same 24 digital roots continue repeating throughout the Fibonacci summation sequence ad infinitum.
Also notice that the first and second half of any Fibonacci sequence of 24 roots digitally compress in respective pairs to the number nine. I.e., the 1st and the 13th digital roots sum together, and that sum then digitally compresses to nine. The 2nd and the 14th, to nine, etc. See Table 1 column, Root of Root Sums.
60-Number repeating pattern in the last digits of the Fibonacci Sequence
Just for the record, I wrote about this wheel of 60 in a published book in 2010, 3 years before Khan, showing the 60 Pattern and the cardinal alignment of the zeroes. It is in one of my 9 books from the series THE BOOK OF PHI, volume 3, sub-titled “The 108 Codes, an Introduction” pages 35 onwards, chapter 3, called Time Code 12:24:60 Encrypted in the Fibonacci Sequence.
“All integers can be represented uniquely as a sum of zero or more "negative" Fibonacci numbers F-1 = 1, F-2 = -1, F-3 = 2, F-4 = -3, provided that no two consecutive elements of this infinite sequence are used. The NegaFibonacci representation leads to an interesting coordinate system for a classic infinite tiling of the hyperbolic plane by triangles, where each triangle has one 90° angle, one 45° angle, and one 36° angle.”
Zeckendorf's theorem, named after Belgian mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers.
Zeckendorf's theorem states that every positive integer can be represented 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.
“4.2. The generalized Fibonacci m-numbers, Metallic Means by Vera Spinadel, Gazale formulas and a general theory of hyperbolic functions. Another generalization of Fibonacci numbers was introduced recently by Vera W. Spinadel , Midchat Gazale , Jay Kappraff  and other scientists. We are talking about the generalized Fibonacci m-numbers that for a given positive real
number m>0 are given by the recursive relation:
Fm(n) = mFm(n-1) + Fm(n-2); Fm(0)=0, Fm(1)=1.(25)
First of all, we note that the recursive relation (25) is reduced to the recursive relation Fn=Fn-1+Fn-2 for the case m=1. For other values of m, the recursive relation (25) generates an infinite number of new recursive numerical sequences.
The following characteristic algebraic equation follows from (25):
x2 – mx – 1 = 0, (26)
which for the case m=1 is reduced to x2=X+1. A positive root of Eq. (26) generates an infinite number of new “harmonic” proportions – “Metallic Means” by Vera Spinadel , which are expressed by the following general formula:
Note that for the case m=1 the formula (27) gives the classical golden mean Φ1 = (1+√5)/2. The metallic means possess the following unique mathematical properties:
which are generalizations of similar properties for the classical golden mean Φ1=Φ(m=1):
Note that the expressions (27), (28) and (29), without doubt, satisfy Dirac’s “Principle of Mathematical
Beauty” and emphasize a fundamental characteristic of both the classical golden mean and the metallic means.
Recently, by studying the recursive relation (25), the Egyptian mathematician Midchat Gazale deduced the following remarkable formula given by Fibonacci m'-numbers:
where m>0 is a given positive real number, Φm is the metallic mean given by (27), n = 0, ±1, ±2, ±3, ....
The author of the present article [Stakhov] named the formula (30) in  formula Gazale for the Fibonacci m-numbers after Midchat Gazale. The similar Gazale formula for the Lucas m-numbers is deduced in
Lm(n) = Φmn + (-1)nΦm-n(31)
It is appropriate to give the following comparative table, which gives a relationship between the
golden mean and metallic means as new mathematical constants of Nature.
A beauty of these formulas is charming. This gives a right to suppose that Dirac’s “Principle of
Mathematical Beauty” can be applicable fully to the metallic means and hyperbolic Fibonacci and
Lucas m-functions. And this, in its turn, gives hope that these mathematical results can become a base
for theoretical natural sciences.
A general theory of hyperbolic functions given by (32)-(35) can lead to the following scientific
theories of a fundamental character: (1) Lobachevski’s “golden” geometry; (2) Minkovski’s
“golden” geometry as an original interpretation of Einstein’s special theory of relativity. In
Lobachevski’s “golden” geometry and Minkovski's "golden" geometry, the processes of the real world
are modeled, in the general case, by the hyperbolic Fibonacci and Lucas m-functions (32)-(35). The
classical Lobachevski geometry, Minkowski geometry and Bodnar geometry  are partial cases of
this general hyperbolic geometry. We propose that this approach is of great importance for
contemporary mathematics and theoretical physics and could become the source of new scientific
It is clear that all mathematical results given by formulas (25)-(45) satisfy Dirac’s “Principle of
Mathematical Beauty.” There can be no doubt that these beautiful mathematical results will be widely
used in modern theoretical natural sciences. Bodnar’s geometry  provides hope of this.”
“The arbitrary three adjacent Fibonacci numbers F(n-1), F(n), F(n+1) (n = 0, +/-1, +/-2, +/-3, …) are connected between themselves with the following mathematical identity:
F2(n) - F(n-1)F(n+1) = (-1)n+1 .(1.5)
The formula (1.5) is called Cassini formula after the French astronomer Giovanni Domenico Cassini (1625-1712) who deduced this formula for the first time.
In 19th century the French mathematician Jacques Philippe Marie Binet (1786-1856) deduced two remarkable formulas, which connect Fibonacci and Lucas numbers with the golden mean:
Note that these formulas were discovered by de Moivre (1667-1754) and Nikolai Bernoulli (1687-1759) a one century before Binet. However, in modern mathematical literature these formulas are called Binet formulas.
"The Fibonacci numbers are Nature's numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind."
“From sunflower seeds to artichoke flowerings, many features in plants follow patterns arranged in terms of Fibonacci numbers: 1, 2, 3, 5, 8, 13… (each number is the sum of the previous two). Research has established that these patterns are optimally packed configurations (of plant organs such as flowers, leaves, or seeds) that maximize access to light and nutrients and thus provide an evolutionary advantage. But how does a plant know how to grow such optimal morphologies? According to a paper in Physical Review Letters, Fibonacci patterns can emerge as a result of the physical and biochemical mechanisms underlying plant growth.
Matthew Pennybacker and Alan C. Newell at the University of Arizona, Tucson, study the case of a sunflower head, in which seeds are grown in annuli surrounding a central region of undifferentiated cells called the meristem—a plant tissue often compared to stem cells in animals. As new seeds are added, the annuli shrink in radius until the head is filled, with the seeds arranged in families of concentric spirals. The authors apply a model that describes how auxin—a growth hormone synthesized by the plant and transported by certain proteins—is distributed within the meristem. The positions in which auxin concentration is maximal indicate where new seeds are formed. Their simulations accurately predict that sunflower seeds form clockwise and counterclockwise spirals, and that the numbers of the two types of spirals are always two consecutive ones in a Fibonacci series. The results suggest that optimal packings may arise in systems described by related families of partial differential equations. – Matteo Rini”
Phyllotaxis, Pushed Pattern-Forming Fronts, and Optimal Packing
Matthew Pennybacker† and Alan C. Newell
Department of Mathematics, University of Arizona, Tucson, Arizona 85721, USA
[Selected for a Synopsis in Physics]
Received 12 December 2012; published 13 June 2013
“We demonstrate that the pattern forming partial differential equation derived from the auxin distribution model proposed by Meyerowitz, Traas, and others gives rise to all spiral phyllotaxis properties observed on plants. We show how the advancing pushed pattern front chooses spiral families enumerated by Fibonacci sequences with all attendant self-similar properties, a new amplitude invariant curve, and connect the results with the optimal packing based algorithms previously used to explain phyllotaxis. Our results allow us to make experimentally testable predictions.”
Fig. 1. Binet's formula curve of real numbers between the Fibonacci sequence integers, from F[-5] through F.
Real, fractional Fibonacci numbers (between the integer numbers) are produced by Binet's analytic formula shown in Fig. 1.
Abraham de Moivre (1667-1754) was the first to work with the mathematics involved, though the formula is named after Jacques Philippe Marie Binet (1786-1856).
The formula in Fig. 1 includes the Golden Ratio and the inverse of the Golden ratio, both using the algebraic form written with the square root of five.
However, the classical form is often seen simplified with the golden ratio substituted by the Greek letter Φ and the inverse 1/Phi denoted as lower case φ. The Binet formula then looks less intimidating with the substitutions of labels:
F[n] =(Φn - (-1/Φ)n ) / √5
F[n] =(Φn - (-φ)n ) / √5
Please be advised —most software interpreters I tried (circa 2010) do not support raising a number to a fractional power. Some did accept the fractional exponent value, but did not throw an exception-error while returning an incorrect value. I found no popular interpreters (Google spreadsheet, Open Office, etc.) that would accept negative values. Most interpreters did throw an error when passed a negative exponent.
Substitute the problematic cyclic -1N term with this identity:
-1N = cos(Nπ)
This avoids the negative-exponent software interpreter bugs, swapping mathematical equivalence by the well behaved cosine function.
Fig. 2. When twisted on the axis of the indices, the Fibonacci real-numbers produced by Binet's Formula create elegant spirals.
Fig. 3. Looking down the twisting axis, orthographically (without perspective) the curves are seen crossing on the integer Fibonacci numbers.
Plotting conjugate Binet's formula curves, 180° out of phase, the deeper negative/positive symmetry is exposed as incoming self-canceling tension as dual opposing vortices in the negative Fibonacci numbers.
Generalized Binet Dynamics
A parameter space image of Binet's Forumla, an analytic function of the Fibonacci sequence. Figure 1 from Generalized Binet Dynamics by Chen Ning and Clifford A. Reiter
The Binet formula provides a mechanism for the Fibonacci numbers to be viewed as a function of a complex variable. The Binet formula may be generalized by using other bases and multiplicative parameters that also give functions of a complex variable. Thus, filled-in Julia sets that exhibit escape time may be constructed. Moreover, these functions have computable critical points and hence we can create escape time images of the critical point based upon the underlying multiplicative parameter. Like the classic Mandelbrot set, these parameter space images provide an atlas of Julia sets.
Journal Computers and Graphics Volume 31 Issue 2, April, 2007, Pages 301-307. Pergamon Press, Inc., Elmsford, NY, USA
Counting musical rythms in ancient India is assumed to be the source of the knowledge of what is now called the Fibonacci Sequence.
Jackie —Spain had 'modern mathematics' two centuries prior to Leonardo of Pisa popularized in numeral mathematics [Liber Abacci 1202].
Fibonacci's book brought modern math to Italy, while Spain was a separated revolution in modernization.
Antoin Lucas named the 'sequence of perfect numbers' 'Fibonacci Sequence', and the nick-name Fibonacci for Leonardo of Pisa from the 1800s.
Kepler, 1600s, hints on vegetable ratios (phyllotaxis).
Post-cards are Golden Rectangles, and many picture frames in the Louvre.
Euclid, book VI: How to divide a line into the extreme and mean ratio?
Plato talks about Euclid's ratio.
Pleasing proportions are near the Golden Ratio. Perhaps we project the ratio in our artifacts without knowledge of the ratio [Jackie].
Platonic solids — tetra- and octa- hedra are described by coordinates of plus/minus one, while the icosa- and dodeca- hedra use Golden ratio coordinates.
Golden Ratio is from ancient Greek geometry and line ratios. Pentagon spiritual figure among Pythagoreans.
Fibonacci numbers mixed in later in Mideval times. Decarte fused geometry and numbers, separate things before him.
Fibonacci numbers are from Arabic and Indian roots, Golden Ratio from ancient Greek [sqrt(5)+1)/2, can be writtne as 5^(.5)*(.5)+(.5) dem].
Patchouli's book, illustrated by Leonardo da Vinci, facial structures, etc, drew Platonic solids as exercise in perspective.
Victorian 1800s —nature and plants realized, tomato has 2 or 3 inside, five outside, cauliflower -5 & 8 flourets, pinapple, 5 & 8 & 13 spirals. Natural solution to a packing problem —0.618 seeds per turn. Gives Fibonacci number of spirals.
Flowers have Fib. numbers and double Fib. number ratios. Four petals are double-two. No flowers have seven.
Vitruvian Man is an animal example [said to not hold Fib. numbers -dem]
Cacti and succulents: 4s, 7s, 11s —Lucas numbers, Fib and Lucas recursive sequences both reveal the Golden Section
Fib. seq. introduces recursive sequences as formula for many sequences.
Any Fib Seq. number squared is one less than the product of the left/right numbers.
Dividable numbers, like 3/6, have Fib. N. at the indices of 3 & 6 that are divisable, 2/8.
Prime index positions are important, acts as markers throughout.
Fib. N. generate Pythagorean triples (e.g. 3,4,5 triangle).
Nature built by simple operations, producing immense complexity —Fib. N. is very simple generator of growth, and rich sequence.
Carducea early 1900s, 'Motula' as a guiding principle in architecture.
Scottsman D.W. Thompson on Growth and Form, 1917, patterns in shapes in nature. Defamation between animal head patterns.
Church in 1910.
Phi, from the name Phideas the Parthenon architect —powers add to give other powers. Nature can do multiplication by addition.
Fibonacci text wasn't published until the 19th century in Italy, no printed editions, just early manuscript copies.
Music —Bartok Cabucci, naturally drawn to Fib. Seq. as a skeleton producing an aesthetic growth and climax.
California —sea shell sections used to produce 30% more efficient rotor.
Liber Abacci now available in an English translation.
"The Fibonacci studies are popular trading tools. Understanding how they are used and to what extent they can be trusted is important to any trader who wants to benefit from the ancient mathematician's scientific legacy."
“The Fibonacci sequence is used as a “hook” to direct interest toward generalizations.”
Keywords: Fibonacci, difference equations, continued fractions, generalized continued fractions, infinite matrix products.
Knuth, D. "Negafibonacci Numbers and the Hyperbolic Plane" Paper presented at the annual meeting of the Mathematical Association of America, The Fairmont Hotel, San Jose, CA [Not Available]. 2010-05-13 from http://www.allacademic.com/meta/p206842_index.html