I used to be a math class representative and I wrote about things that are not difficult, such as the study of the Fibonacci sequence

The Fibonacci sequence,

and The golden section sequence refers to such a sequence: 1, 1, 2, 3, 5, 8, 13, 21,... In mathematics, the Fibonacci sequence is recursively defined as follows: F0= 0, F1=1, Fn=F(n-1)+F(n-2) (n>=2, n∈N*) In the fields of modern physics, quasicrystal structure, chemistry and other fields, the Fibonacci sequence is It has direct application. To this end, the American Mathematical Society has published a mathematics magazine named "Fibonacci Sequence Quarterly" since 1963 to specifically publish research results in this area.

Definition

The Fibonacci sequence refers to the sequence 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

Specially pointed out: item 0 is 0, item 1 is the first 1.

This sequence starts with the second term, and each term is equal to the sum of the previous two terms.

The inventor of the Fibonacci sequence was the Italian mathematician Leonardo Fibonacci

Recursion formula

Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

If F(n) is The nth item of the sequence (n∈N*), then this sentence can be written in the following form:

Obviously this is a linear recursion sequence.

General formula

(As above, also known as the "ratio formula", it is an example of using irrational numbers to represent rational numbers.

)

Note: at this time a1=1, a2=1, an=a(n-1)+a(n-2) (n>=3, n∈N*)

Derivation of the general formula

Method 1: Using the characteristic equation (linear algebra solution)

The characteristic equation of the linear recursion sequence is:

X ^2=X+1

Solution

X1=(1+√5)/2, X2=(1-√5)/2.

Then F(n)=C1*X1^n + C2*X2^n

∵F(1)=F(2)=1

∴C1*X1 + C2 *X2=C1*X1^2 + C2*X2^2=1　

The solution is C1=1/√5, C2=-1/√5

∴F(n )=(1/√5)*{n+Fn=Fn, Fn-Fn=Fn=F(n-1),

n

1

3

4

5

6

7

8

9

10

…

Fn

1

4< /p>

5

9

14

23

37

60

97

157

…

Fn

1

3

7

11

18

29

47

76

123

…

Fn-Fn

1

1

2

3

5

8

13

21

34< /p>

…

Fn+Fn

2

7

9

16< /p>

25

41

66

107

173

280

…

②Any Fibonacci-Lucas sequence can be obtained by the sum of finite terms of the Fibonacci sequence, such as

n

1

2

3

4

5

6

7

8

9

10

…

F(n)

1

1

2

3

5

8

21

34

55

…

F(n-1)

1

1

2

3

5

8

13

21

34

…

F(n-1)

1< /p>

1

2

3

5

8

13

21

34

…

Fn

1

3

7

11

18

29

47

76

123

…

Golden Characteristics and Twin Fibonacci-Lucas Sequences

Fibonacci-Lu Another identical property of the Cass sequence: the absolute value of the difference between the square of the middle term and the product of the two preceding and following terms is a constant value,

Fibonacci sequence: |1*1 -1*2|=|2*2-1*3|=|3*3-2*5|=|5*5-3*8|=|8*8-5*13|=…=1< /p>

Lucas sequence: |3*3-1*4|=|4*4-3*7|=…=5

F[1, 4] sequence: | 4*4-1*5|=11

F[2,5] sequence: |5*5-2*7|=11

F[2,7] sequence : |7*7-2*9|=31

The value of the Fibonacci sequence is 1 as the minimum, which means that the ratio of the front and back terms is close to the golden ratio the fastest. We call it the golden characteristic. The golden characteristic The only number sequence of 1 is the Fibonacci sequence, which is a unique sequence. The golden feature of the Lucas sequence is 5, which is also a unique sequence. The only two unique number sequences that are relatively prime are the Fibonacci sequence and the Lucas sequence.

The golden characteristics of F[1, 4] and F[2, 5] are both 11, which is a twin sequence.

F[2,7] also has a twin sequence: F[3,8]. The other first two Fibonacci-Lucas sequences that are relatively prime are twin sequences, called twin Fibonacci-Lucas sequences.

Generalized Fibonacci Sequence

The golden feature of Fibonacci Sequence 1 also reminds us of the Pell Sequence: 1, 2, 5, 12, 29,… , there is also |2*2-1*5|=|5*5-2*12|=...=1 (this eigenvalue of this type of sequence is called Pythagorean characteristic).

The recursion rule of Pell sequence Pn: P1=1, P2=2, Pn=P(n-2)+2P(n-1).

And so on. The general rule for deriving the third term from the first two terms: f(n) = f(n-1) * p + f(n-2) * q, is called the generalized Fibonacci sequence.

When p=1, q=1, we get the Fibonacci-Lucas sequence.

When p=1, q=2, we get the Pell-Pythagorean number (a set of sequences related to right triangles with integer side lengths).

When p=-1, q=2, we get an arithmetic sequence. When f1=1 and f2=2, we get the natural number sequence 1, 2, 3, 4…. The characteristic of a natural number sequence is that the difference between the square of each number and the product of the two numbers before and after it is 1 (this difference in an arithmetic sequence is called a natural characteristic).

A generalized Fibonacci sequence p=±1 that is similar to gold characteristics, Pythagorean characteristics, and natural characteristics.

When f1=1, f2=2, p=2, q=1, we get the geometric sequence 1, 2, 4, 8, 16...

Related mathematics

Permutation and combination

There is a staircase with 10 steps. It is stipulated that each step can only cross one or two steps. There are several different ways to climb the 10th step. ?

This is a Fibonacci sequence: there is one way to climb the first step; there are two ways to climb the two steps; there are three ways to climb the third step. ; There are five ways to climb the four steps...

1, 2, 3, 5, 8, 13... So, there are 89 ways to climb the tenth step.

Similarly, if an even coin is tossed 10 times, how many possible situations are there in which heads do not appear continuously?

The answer is (1/√5)*{[(1+√5)/2]^(12) - [(1-√5)/2]^(12) }=144 species.

Find the general formula of the recursive sequence a⑴=1, a(n+1)=1+1/a(n)

By mathematical induction, we can get: a( n)=F(n+1)/F(n), substitute the general expression of the Fibonacci sequence and simplify to get the result.

Rabbit breeding problem

The Fibonacci sequence was introduced by mathematician Leonardo Fibonacci using rabbit breeding as an example, so it is also called the "rabbit sequence" ".

Generally speaking, rabbits have the ability to reproduce two months after they are born. A pair of rabbits can give birth to a pair of baby rabbits every month. If all the rabbits do not die, how many pairs of rabbits will be produced after one year?

We might as well analyze a pair of newly born bunnies:

The bunnies have no reproductive ability in the first month, so they are still a pair

Two Three months later, a pair of little rabbits were born, and there were two pairs in total.

Three months later, the old rabbit gave birth to another pair. Because the little rabbits were not yet capable of reproduction, one pair*** There are three pairs

－－－－－－

By analogy, the following table can be listed:

Number of months passed

1< /p>

2

3

4

5

6

7

8

9

10

11

12

Pairs of cubs

1

1

1

2

3

5

8

13

21

34

55

89

Adult rabbit logarithm

1

1

2

3

5

13

21

34

55

89

144

Population logarithm

1

1

2

3

5

8

13

21

34

55

89< /p>

144

233

The number of pairs of pups = the number of pairs of adult rabbits in the previous month

The number of pairs of adult rabbits = the number of pairs of adult rabbits in the previous month + Number of pairs of cubs in the previous month

Overall number of pairs = Number of pairs of adult rabbits this month + Number of pairs of cubs this month

It can be seen that number of pairs of cubs, number of adult rabbits, and overall number Logarithms form a sequence.

There is a very obvious feature about this sequence, that is: the sum of the two adjacent items in the front constitutes the last item.

This sequence was proposed by the Italian medieval mathematician Fibonacci in "Abacus Encyclopedia". The general formula of this series, in addition to having a(n+2)=an+a(n+ In addition to the properties of 1), it can also be proved that the general formula is: an=(1/√5)*{[(1+√5)/2]^n-[(1-√5)/2]^n} (n=1,2,3....)

Sequences and matrices

For the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13 ,……. There is the following definition

F(n)=f(n-1)+f(n-2)

F(1)=1

F( 2)=1

For the following matrix multiplication

F(n+1) = 11 F(n)

F(n) 10 F(n- 1)

The operation is to multiply the matrix 11 on the right by the matrix F(n) to get:

10 F(n-1)

F(n +1)=F(n)+F(n-1)

F(n)=F(n)

It can be seen that the multiplication of this matrix is ??completely consistent with the Fibonacci sequence Definition

Assume matrix A=1 1 and iterate n times to get: F(n+1) =A^(n) * F(1)= A^(n)*1

1 0 F(n) F(0) 0

This is the matrix multiplication definition of the Fibonacci sequence.

Another algorithm for matrix multiplication is A^n (n is an even number) = A^(n/2)* A^(n/2). In this way, we can realize logarithms through the idea of ??bisection. Complexity of matrix multiplication.

So the answer can be obtained recursively.

Another way to find sequence values:

F(n) = [ (( sqrt ( 5 ) + 1 ) / 2) ^ n ]

Where [x] means taking the nearest integer from x.

Fibonacci Arc

Fibonacci arc, also known as Fibonacci fan line. First, the trend line is drawn based on two endpoints, for example, the two points on the line from the lowest point to the highest point. Then draw an "invisible" vertical line through the second point. Then, draw a third trend line from the first point: the intersection of the invisible vertical lines at 38.2%, 50%, and 61.8%.

Fibonacci arcs are potential support and resistance level price levels. Fibonacci Arcs and Fibonacci Fans are often drawn simultaneously on charts. Support and resistance points are derived from the intersection of these lines.

It is important to note that the intersection point of the arc and the price curve will change depending on the value range of the chart, because the arc is part of the circle and its formation is always the same.

He died in 1170 AD and his native place was Pisa. He was known as the "Leonardo of Pisa". In 1202, he wrote Liber Abacci. He was the first European to study Indian and Arabic mathematical theories. His father was employed as a diplomatic consul by a business group in Pisa, in an area equivalent to today's Algeria, so Leonardo was able to study mathematics under the guidance of an Arab teacher. He also studied mathematics in Egypt, Syria, Greece, Sicily and Provence.

Application of Fibonacci Sequence in the Stock Market

Time cycle theory is one of the fundamental reasons for the rise and fall of stock prices, and it can explain the mysteries of most market rises and falls. In the time cycle cycle theory, in addition to using fixed time cycle numbers to find variable inventory points, the relationship between bands can also be used for research. But no matter how you look for change points, the Fibonacci sequence is one of the foundations of various important analyses. This article will briefly explain the Fibonacci sequence and its relationship with the market.

Tools/Materials

Steps/Methods

The Fibonacci sequence was discovered by the Italian mathematician Fibonacci in the 13th century. A series of numbers in a sequence are often called magic numbers. The specific numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, etc. Starting from the third number of the sequence, each number is equal to the previous two adjacent numbers. The sum of numbers. The quotient of two adjacent items in the Fibonacci sequence is close to the golden section number 0.618. The numbers related to this number, such as 0.191, 0.382, 0.5 and 0.809, constitute important numbers in the stock market regarding market time and space calculations.

Fibonacci numbers can be found in the laws of various phenomena ranging from the entire universe to molecules and atoms, from time to space, from nature to human society, politics, economy, military, etc. traces. The shadow of 0.618 can be found in world-famous buildings such as Notre Dame de Paris, Eiffel Tower, Egyptian Pyramids, etc. The themes of famous paintings, photography, sculptures and other works are all located at 0.618 of the painting. The announcer stands at 0.618 on the stage and gives the sweetest and most beautiful voice.

The human belly button is 0.618 of the length of the human body, and the human knee is 0.618 of the length from the sole of the foot to the navel. The use of 0.618 in war is also ubiquitous, from the manufacture of weapons, to the deployment of troops, to the use of war time cycles. It is said that Napoleon the Great was defeated by the Golden Section.

In the analysis methods of financial markets, Fibonacci numbers appear frequently. For example, in the wave theory, a bull market can be represented by 1 rising wave, or 5 small waves at a lower level, and can be further subdivided into 21 or 89 small waves; in the spatial analysis system Among them, the height of the rebound market is usually 0.382, 0.5, and 0.618 of the previous downward trend; the callback market is usually 0.382, 0.5, and 0.618 of the previous upward trend.

The Fibonacci sequence has two important meanings in actual operation:

The first practical significance lies in the sequence itself. The dozen numbers in front of this sequence play an important role in the time relationship of the market's daily line. When the market is in an important key change time area, these numbers can determine the specific change time. When using the Fibonacci sequence, you can calculate from an important stage in the market to the future market. When the time is reached, the probability of the market changing direction is high.

Figure 1 Composite Index (1A0001) Daily Chart from July 29 to December 31, 2009

As shown in Figure 1, Composite Index (1A0001) August 2009 The time relationship from 3478 points on the 4th to the stage low of 2639 points on September 1, 2009 is 21 trading days. The stage low of 2639 points on September 1, 2009 to the high point of 3068 points on September 18, 2009 It takes 13 trading days. It takes 21 trading days to reach the low of 2712 points on September 29, 2009. It takes 34 trading days to reach the high of 3123 points on October 23, 2009. It takes 34 trading days to reach the low of 2712 points on September 29, 2009. The annual second high of 3361 points on March 24 was 55 trading days.

Figure 2 Composite Index (1A0001) Weekly Chart from July 10 to December 31, 2009

As shown in Figure 2, Composite Index (1A0001) August 2009 The running time from the high point of 3478 points on the 4th to 2639 points on September 4, 2009 is 5 weeks; the time from the low point of 2639 points on September 4, 2009 to the rebound high point of 3361 points on November 27, 2009 is 13 week.

The application of Fibonacci numbers in the stock market

The application of Fibonacci numbers in the stock market

The second practical significance lies in the Derived numbers are the theoretical basis for calculating future market changes in vertical time cycles in the market. This set of derivative series are: 1.236, 1.309, 1.5, 1.618, 1.809, 2, 2.236, 2.382, 2.5 and a series of numbers related to the golden section 0.618.

There are six important time calculation methods when using the magic sequence:

First, calculate the running time of the rising wave band in the future market through the complete falling wave period.

Second, calculate the running time of the downtrend in the future market through the complete uptrend period.

These two proportional relationships are like the relationship between action and reaction that we often see in life. The height of a table tennis ball falling vertically to the ground determines the height of the table tennis ball rebounding after hitting the ground. .

Third, calculate the final running time of this rising band through the time from the low point to the high point of the first sub-band in the rising band.

Fourth, calculate the final running time of this down band through the time from the high point to the low point of the first sub-band in the down band.

These two proportional relationships are like the relationship between driving force and inertia that we often see in life. When the distance between the bow and the string of an ancient bow and arrow is stretched, it directly determines the forward distance of the arrow in the future. .

Fifth, calculate the final running time of the future rising band through the time of the two adjacent lows of the first sub-band in this rising band.

Sixth, calculate the final running time of this down band through the time of the two adjacent high points of the first sub-band in the down band.

These two proportional relationships are as important as the width of the building foundation that we often see in life affects the future height. In the case of the same material, the greater the width of the foundation, the higher the future height.

5

The most important of these six important time calculation methods is the parameters actually used in the calculation process. Using different parameters will get different answers, and the use process Almost all important parameters in are related to the Fibonacci sequence. Due to space reasons, I will leave a foreshadowing here. I will explain the calculation method in detail for stock investors in future articles.