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Fibonacci Sequence

The Fibonacci Sequence is the series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, .

The next number is found by adding up the two numbers before it.

  • The 2 is found by adding the two numbers before it (1+1)
  • The 3 is found by adding the two numbers before it (1+2),
  • And the 5 is (2+3),
  • and so on!

Example: the next number in the sequence above is 21+34 = 55

It is that simple!

Here is a longer list:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, .

Can you figure out the next few numbers?

Makes A Spiral

When we make squares with those widths, we get a nice spiral:

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Do you see how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.

The Rule

The Fibonacci Sequence can be written as a „Rule“ (see Sequences and Series).

First, the terms are numbered from 0 onwards like this:

n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 .
xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 .

So term number 6 is called x6 (which equals 8).

Example: the 8th term is
the 7th term plus the 6th term:

So we can write the rule:

Example: term 9 is calculated like this:

Golden Ratio

And here is a surprise. When we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio „φ“ which is approximately 1.618034.

In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:

1.666666666. 1.618055556. 1.618025751.

Note: this also works when we pick two random whole numbers to begin the sequence, such as 192 and 16 (we get the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, . ):

It takes longer to get good values, but it shows that not just the Fibonacci Sequence can do this!

Using The Golden Ratio to Calculate Fibonacci Numbers

And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:

The answer always comes out as a whole number, exactly equal to the addition of the previous two terms.

When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer 8.00000033. A more accurate calculation would be closer to 8.

Try it for yourself!

You can also calculate a Fibonacci Number by multiplying the previous Fibonacci Number by the Golden Ratio and then rounding (works for numbers above 1):

Example: 8 × φ = 8 × 1.618034. = 12.94427. = 13 (rounded)

Some Interesting Things

Here is the Fibonacci sequence again:

n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 .
xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 .

There is an interesting pattern:

  • Look at the number x3 = 2. Every 3rd number is a multiple of 2 (2, 8, 34, 144, 610, . )
  • Look at the number x4 = 3. Every 4th number is a multiple of 3 (3, 21, 144, . )
  • Look at the number x5 = 5. Every 5th number is a multiple of 5 (5, 55, 610, . )

And so on (every nth number is a multiple of xn).

1/89 = 0.011235955056179775.

Notice the first few digits (0,1,1,2,3,5) are the Fibonacci sequence?

In a way they all are, except multiple digit numbers (13, 21, etc) overlap, like this:

. etc .
0.011235955056179775. = 1/89

Terms Below Zero

The sequence works below zero also, like this:

n = . -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 .
xn = . -8 5 -3 2 -1 1 0 1 1 2 3 5 8 .

(Prove to yourself that each number is found by adding up the two numbers before it!)

In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a +-+- . pattern. It can be written like this:

Which says that term „-n“ is equal to (в€’1) n+1 times term „n“, and the value (в€’1) n+1 neatly makes the correct 1,-1,1,-1. pattern.


Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!

About Fibonacci The Man

His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.

„Fibonacci“ was his nickname, which roughly means „Son of Bonacci“.

As well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic Numerals (like our present numbers 0,1,2,3,4,5,6,7,8,9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo.

Fibonacci Day

Fibonacci Day is November 23rd, as it has the digits „1, 1, 2, 3“ which is part of the sequence. So next Nov 23 let everyone know!


Fibonacci, also called Leonardo Pisano, English Leonardo of Pisa, original name Leonardo Fibonacci, (born c. 1170, Pisa?—died after 1240), medieval Italian mathematician who wrote Liber abaci (1202; “Book of the Abacus”), the first European work on Indian and Arabian mathematics.

Little is known about Fibonacci’s life beyond the few facts given in his mathematical writings. During Fibonacci’s boyhood his father, Guglielmo, a Pisan merchant, was appointed consul over the community of Pisan merchants in the North African port of Bugia (now Bejaïa, Algeria). Fibonacci was sent to study calculation with an Arab master. He later went to Egypt, Syria, Greece, Sicily, and Provence, where he studied different numerical systems and methods of calculation.

When Fibonacci’s Liber abaci first appeared, Hindu-Arabic numerals were known to only a few European intellectuals through translations of the writings of the 9th-century Arab mathematician al-Khwārizmī. The first seven chapters dealt with the notation, explaining the principle of place value, by which the position of a figure determines whether it is a unit, 10, 100, and so forth, and demonstrating the use of the numerals in arithmetical operations. The techniques were then applied to such practical problems as profit margin, barter, money changing, conversion of weights and measures, partnerships, and interest. Most of the work was devoted to speculative mathematics— proportion (represented by such popular medieval techniques as the Rule of Three and the Rule of Five, which are rule-of-thumb methods of finding proportions), the Rule of False Position (a method by which a problem is worked out by a false assumption, then corrected by proportion), extraction of roots, and the properties of numbers, concluding with some geometry and algebra. In 1220 Fibonacci produced a brief work, the Practica geometriae (“Practice of Geometry”), which included eight chapters of theorems based on Euclid’s Elements and On Divisions.

The Liber abaci, which was widely copied and imitated, drew the attention of the Holy Roman emperor Frederick II. In the 1220s Fibonacci was invited to appear before the emperor at Pisa, and there John of Palermo, a member of Frederick’s scientific entourage, propounded a series of problems, three of which Fibonacci presented in his books. The first two belonged to a favourite Arabic type, the indeterminate, which had been developed by the 3rd-century Greek mathematician Diophantus. This was an equation with two or more unknowns for which the solution must be in rational numbers (whole numbers or common fractions). The third problem was a third-degree equation (i.e., containing a cube), x 3 + 2x 2 + 10x = 20 (expressed in modern algebraic notation), which Fibonacci solved by a trial-and-error method known as approximation; he arrived at the answer

in sexagesimal fractions (a fraction using the Babylonian number system that had a base of 60), which, when translated into modern decimals (1.3688081075), is correct to nine decimal places.

Contributions to number theory

For several years Fibonacci corresponded with Frederick II and his scholars, exchanging problems with them. He dedicated his Liber quadratorum (1225; “Book of Square Numbers”) to Frederick. Devoted entirely to Diophantine equations of the second degree (i.e., containing squares), the Liber quadratorum is considered Fibonacci’s masterpiece. It is a systematically arranged collection of theorems, many invented by the author, who used his own proofs to work out general solutions. Probably his most creative work was in congruent numbers—numbers that give the same remainder when divided by a given number. He worked out an original solution for finding a number that, when added to or subtracted from a square number, leaves a square number. His statement that x 2 + y 2 and x 2 − y 2 could not both be squares was of great importance to the determination of the area of rational right triangles. Although the Liber abaci was more influential and broader in scope, the Liber quadratorum alone ranks Fibonacci as the major contributor to number theory between Diophantus and the 17th-century French mathematician Pierre de Fermat.

Except for his role in spreading the use of the Hindu-Arabic numerals, Fibonacci’s contribution to mathematics has been largely overlooked. His name is known to modern mathematicians mainly because of the Fibonacci sequence (see below) derived from a problem in the Liber abaci:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

The resulting number sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 (Fibonacci himself omitted the first term), in which each number is the sum of the two preceding numbers, is the first recursive number sequence (in which the relation between two or more successive terms can be expressed by a formula) known in Europe. Terms in the sequence were stated in a formula by the French-born mathematician Albert Girard in 1634: un + 2 = un + 1 + un, in which u represents the term and the subscript its rank in the sequence. The mathematician Robert Simson at the University of Glasgow in 1753 noted that, as the numbers increased in magnitude, the ratio between succeeding numbers approached the number α, the golden ratio, whose value is 1.6180…, or (1 + Square root of √ 5 )/2. In the 19th century the term Fibonacci sequence was coined by the French mathematician Edouard Lucas, and scientists began to discover such sequences in nature; for example, in the spirals of sunflower heads, in pine cones, in the regular descent (genealogy) of the male bee, in the related logarithmic (equiangular) spiral in snail shells, in the arrangement of leaf buds on a stem, and in animal horns.

5 способов вычисления чисел Фибоначчи: реализация и сравнение


Программистам числа Фибоначчи должны уже поднадоесть. Примеры их вычисления используются везде. Всё от того, что эти числа предоставляют простейший пример рекурсии. А ещё они являются хорошим примером динамического программирования. Но надо ли вычислять их так в реальном проекте? Не надо. Ни рекурсия, ни динамическое программирование не являются идеальными вариантами. И не замкнутая формула, использующая числа с плавающей запятой. Сейчас я расскажу, как правильно. Но сначала пройдёмся по всем известным вариантам решения.

Код предназначен для Python 3, хотя должен идти и на Python 2.

Для начала – напомню определение:

Замкнутая формула

Пропустим детали, но желающие могут ознакомиться с выводом формулы. Идея в том, чтобы предположить, что есть некий x, для которого Fn = x n , а затем найти x.

Решаем квадратное уравнение:

Откуда и растёт «золотое сечение» ϕ=(1+√5)/2. Подставив исходные значения и проделав ещё вычисления, мы получаем:

что и используем для вычисления Fn.

Быстро и просто для малых n
Требуются операции с плавающей запятой. Для больших n потребуется большая точность.
Использование комплексных чисел для вычисления Fn красиво с математической точки зрения, но уродливо — с компьютерной.


Самое очевидное решение, которое вы уже много раз видели – скорее всего, в качестве примера того, что такое рекурсия. Повторю его ещё раз, для полноты. В Python её можно записать в одну строку:

Очень простая реализация, повторяющая математическое определение
Экспоненциальное время выполнения. Для больших n очень медленно
Переполнение стека


У решения с рекурсией есть большая проблема: пересекающиеся вычисления. Когда вызывается fib(n), то подсчитываются fib(n-1) и fib(n-2). Но когда считается fib(n-1), она снова независимо подсчитает fib(n-2) – то есть, fib(n-2) подсчитается дважды. Если продолжить рассуждения, будет видно, что fib(n-3) будет подсчитана трижды, и т.д. Слишком много пересечений.

Поэтому надо просто запоминать результаты, чтобы не подсчитывать их снова. Время и память у этого решения расходуются линейным образом. В решении я использую словарь, но можно было бы использовать и простой массив.

(В Python это можно также сделать при помощи декоратора, functools.lru_cache.)

Просто превратить рекурсию в решение с запоминанием. Превращает экспоненциальное время выполнение в линейное, для чего тратит больше памяти.
Тратит много памяти
Возможно переполнение стека, как и у рекурсии

Динамическое программирование

После решения с запоминанием становится понятно, что нам нужны не все предыдущие результаты, а только два последних. Кроме этого, вместо того, чтобы начинать с fib(n) и идти назад, можно начать с fib(0) и идти вперёд. У следующего кода линейное время выполнение, а использование памяти – фиксированное. На практике скорость решения будет ещё выше, поскольку тут отсутствуют рекурсивные вызовы функций и связанная с этим работа. И код выглядит проще.

Это решение часто приводится в качестве примера динамического программирования.

Быстро работает для малых n, простой код
Всё ещё линейное время выполнения
Да особо ничего.

Матричная алгебра

И, наконец, наименее освещаемое, но наиболее правильное решение, грамотно использующее как время, так и память. Его также можно расширить на любую гомогенную линейную последовательность. Идея в использовании матриц. Достаточно просто видеть, что

А обобщение этого говорит о том, что

Два значения для x, полученных нами ранее, из которых одно представляло собою золотое сечение, являются собственными значениями матрицы. Поэтому, ещё одним способом вывода замкнутой формулы является использование матричного уравнения и линейной алгебры.

Так чем же полезна такая формулировка? Тем, что возведение в степень можно произвести за логарифмическое время. Это делается через возведения в квадрат. Суть в том, что

где первое выражение используется для чётных A, второе для нечётных. Осталось только организовать перемножения матриц, и всё готово. Получается следующий код. Я организовал рекурсивную реализацию pow, поскольку её проще понять. Итеративную версию смотрите тут.

Фиксированный объём памяти, логарифмическое время
Код посложнее
Приходится работать с матрицами, хотя они не так уж и плохи

Сравнение быстродействия

Сравнивать стоит только вариант динамического программирования и матрицы. Если сравнивать их по количеству знаков в числе n, то получится, что матричное решение линейно, а решение с динамическим программированием – экспоненциально. Практический пример – вычисление fib(10 ** 6), числа, у которого будет больше двухсот тысяч знаков.

n = 10 ** 6
Вычисляем fib_matrix: у fib(n) всего 208988 цифр, расчёт занял 0.24993 секунд.
Вычисляем fib_dynamic: у fib(n) всего 208988 цифр, расчёт занял 11.83377 секунд.

Теоретические замечания

Не напрямую касаясь приведённого выше кода, данное замечание всё-таки имеет определённый интерес. Рассмотрим следующий граф:

Подсчитаем количество путей длины n от A до B. Например, для n = 1 у нас есть один путь, 1. Для n = 2 у нас опять есть один путь, 01. Для n = 3 у нас есть два пути, 001 и 101. Довольно просто можно показать, что количество путей длины n от А до В равно в точности Fn. Записав матрицу смежности для графа, мы получим такую же матрицу, которая была описана выше. Это известный результат из теории графов, что при заданной матрице смежности А, вхождения в А n — это количество путей длины n в графе (одна из задач, упоминавшихся в фильме «Умница Уилл Хантинг»).

Почему на рёбрах стоят такие обозначения? Оказывается, что при рассмотрении бесконечной последовательности символов на бесконечной в обе стороны последовательности путей на графе, вы получите нечто под названием „подсдвиги конечного типа“, представляющее собой тип системы символической динамики. Конкретно этот подсдвиг конечного типа известен, как «сдвиг золотого сечения», и задаётся набором «запрещённых слов» <11>. Иными словами, мы получим бесконечные в обе стороны двоичные последовательности и никакие пары из них не будут смежными. Топологическая энтропия этой динамической системы равна золотому сечению ϕ. Интересно, как это число периодически появляется в разных областях математики.

Fibonacci Retracement Definition & Levels

What is a Fibonacci Retracement?

A Fibonacci retracement is a term used in technical analysis that refers to areas of support or resistance. Fibonacci retracement levels use horizontal lines to indicate where possible support and resistance levels are. Each level is associated with a percentage. The percentage is how much of a prior move the price has retraced. The Fibonacci retracement levels are 23.6%, 38.2%, 61.8% and 78.6%. While not officially a Fibonacci ratio, 50% is also used.

The indicator is useful because it can be drawn between any two significant price points, such as a high and a low, and then the indicator will create the levels between those two points.

If the price rises $10, and then drops $2.36, it has retraced 23.6%, which is a Fibonacci number. Fibonacci numbers are found throughout nature, and therefore many traders believe that these numbers also have relevance in the financial markets.

Key Takeaways

  • The indicator connects any two points that the trader views at relevant, typically a high and low point.
  • Once the indicator has been drawn on the chart, the levels are fixed and will not change. The percentage levels provided are areas where the price could stall or reverse.
  • Levels should not be relied on exclusively. For example, it is dangerous to assume the price will reverse after hitting a specific Fibonacci level. It may, but it also may not.
  • Fibonacci retracement levels are most frequently used to provide potential areas of interest. If a trader wants to buy, they watch for the price to stall at a Fibonacci level and then bounce off that level before buying.
  • The most commonly used ratios include 23.6%, 38.2%, 50%, 61.8% and 78.6%. These represent how much of a prior move the price has corrected or retraced.

The Formulas for Fibonacci Retracement Levels Are:

The indicator itself doesn’t have any formulas. When the indicator is applied to a chart the user chooses two points. Once those two points are chosen, the lines are drawn at percentages of that move.

If the price rises from $10 to $15, and these two prices levels are the points used to draw the retracement indicator, then 23.6% level will be at $13.82 ($15 – ($5 x 0.236)) = $13.82. The 50% level will be at $12.50 ($15 – ($5 x 0.5)) = $12.50.

How to Calculate Fibonacci Retracement Levels

As discussed above, there is nothing to calculate when it comes to Fibonacci retracement levels. They are simply percentages of whatever price range is chosen.

You may wonder where these numbers come from, though. They are based on something called the Golden Ratio.

If you start a sequence of numbers with zero and one, and then keep adding the prior two numbers, you end up with a number string like this:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987. with the string continuing on indefinitely.

The Fibonacci retracement levels are all derived from this number string. Excluding the first few numbers, as the sequence gets going, if you divide one number by the next number you get 0.618, or 61.8%. Divide a number by the second number to its right and you get 0.382 or 38.2%. All the ratios, except for 50% since it is not an official Fibonacci number, are based on some mathematical calculation involving this number string.

Interestingly, the Golden Ratio of 0.618 or 1.618 is found in sunflowers, galaxy formations, shells, historical artifacts and architecture.

Fibonacci Retracement

What Do Fibonacci Retracement Levels Tell You?

Fibonacci retracements can be used to place entry orders, determine stop loss levels, or set price targets. For example, a trader may see a stock moving higher. After a move up it retraces to the 61.8%% level, and then starts to bounce again. Since the bounce occurred at a Fibonacci level, and the longer trend is up, the trader decides to buy. They could set a stop loss at the 78.6% level, or the 100% level (where the move started).

Fibonacci levels are used in other forms technical analysis as well. For example, they are prevalent in Gartley patterns and Elliott Wave theory. After a significant price movement up or down, when the price retraces (which it always does), these forms of technical analysis find the retracements will tend to reverse near certain Fibonacci levels.

Fibonacci retracement levels are static prices that do not change, unlike moving averages. The static nature of the price levels allows for quick and easy identification. This allows traders and investors to anticipate and react prudently when the price levels are tested. These levels are inflection points where some type of price action is expected, either a rejection or a break.

The Difference Between Fibonacci Retracements and Fibonacci Extensions

While Fibonacci retracements apply percentages to a pullback, Fibonacci extensions apply percentages to a move back in the trending direction. For example, a stock goes from $5 to $10, and then back to $7.50. The move from $10 to $7.50 is a retracement. If the price starts rallying again and goes to $16, that is an extension.

Limitations of Using Fibonacci Retracement Levels

While the retracement levels indicate where the price could potentially find support or resistance, there are no assurances the price will actually stop there. This is why other confirmation signals are often used, such as the price actually starting to bounce off the level.

The other argument against Fibonacci retracement levels is that there are so many of them that the price is likely to reverse near one of them quite often. The problem is that in advance traders struggle to know which one will be useful on the current retracement they are analyzing.

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