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Fibonacci series,

Also known as golden section series, it refers to such a series: 1, 1, 2, 3, 5, 8, 13, 2 1, ... Mathematically, Fibonacci series is defined recursively as follows: F0=0, F65438+. =2, n∈N*) Fibonacci sequence has direct applications in modern physics, quasicrystal structure, chemistry and other fields. Therefore, the American Mathematical Society published a mathematical magazine named Fibonacci Series Quarterly from 1963 to publish the research results in this field.

definition

Fibonacci series refers to 0, 1, 1, 2, 3, 5, 8, 13, 2 1, 34, 55, 89, 144, 233, 377.

Specifically, item 0 is 0, and item 1 is item 1.

This series begins with the second term, and each term is equal to the sum of the first two terms.

The inventor of Fibonacci sequence is Italian mathematician Leonardo Fibonacci.

recurrence formula

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

If F(n) is the nth term of the series (n∈N*), then this sentence can be written as follows:

Obviously, this is a linear recursive sequence.

General term formula

(As mentioned above, it is also called "Binet.Alfred formula", which is an example of using irrational numbers to represent rational numbers. )

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

Derivation of general term formula

Method 1: The characteristic equation (linear algebraic solution) was used.

The characteristic equation of linear recursive sequence is:

X^2=X+ 1

solve

X 1=( 1+√5)/2，X2=( 1-√5)/2。

Then f (n) = c1* x1n+C2 * x2n.

∫F( 1)= F(2)= 1

∴c 1*x 1+c2*x2=c 1*x 1^2+c2*x2^2= 1

The solution is c1=1√ 5, C2 =-1√ 5.

∴f(n)=( 1/√5)*{n+fn=fn,fn-fn=f[0, 1]n=f[ 1, 1](n- 1)，

n

1

2

three

four

five

six

seven

eight

nine

10

…

[Mathematics] Function

1

four

five

nine

14

23

37

60

97

157

…

[Mathematics] Function

1

three

four

seven

1 1

18

29

47

76

123

…

Fn-Fn

1

1

2

three

five

eight

13

2 1

34

…

Fn+Fn

2

seven

nine

16

25

4 1

66

107

173

280

…

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

n

1

2

three

four

five

six

seven

eight

nine

10

…

F[ 1， 1](n)

1

1

2

three

five

eight

13

2 1

34

55

…

F[ 1， 1](n- 1)

1

1

2

three

five

eight

13

2 1

34

…

F[ 1， 1](n- 1)

1

1

2

three

five

eight

13

2 1

34

…

[Mathematics] Function

1

three

four

seven

1 1

18

29

47

76

123

…

Golden feature and twin Fibonacci-Lucas sequence

Another homomorphism of Fibonacci-Lucas sequence: the absolute value of the difference between the square of the middle term and the product of the first two terms is a constant value,

Fibonacci series: |1*1-1* 2 | = | 2 * 2-1* 3 | = | 3 * 3-2 * 5 | = | 5 * 3 * 8 | = | 8 *

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

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

F [2 2,5] series: |5*5-2*7|= 1 1

F [2 2,7] series: |7*7-2*9|=3 1

Fibonacci series has the minimum value of 1, that is, the ratio of the front and rear terms is close to the golden section ratio, which is the fastest. We call it the golden feature, and the golden feature sequence of 1 is only Fibonacci sequence, which is the only sequence. The golden feature of Lucas sequence is 5, which is also the only child sequence. The first two series with only coprime are Fibonacci series and Lucas series.

The golden characteristics of F [1, 4] and f [2,5] are both 1 1, which are twin sequences. F [2,7] also has a twin sequence: F [3,8]. The other two coprime Fibonacci-Lucas sequences are twin sequences, which are called twin Fibonacci-Lucas sequences.

Generalized Fibonacci sequence

The golden characteristic of Fibonacci sequence is 1, which reminds us of Pell sequence: 1, 2,5,12,29, …, and | 2 * 2-1* 5 | = | 5 * 2 */kloc-.

The recurrence rules of Pell sequence Pn are: P 1= 1, P2=2 = p (n-2)+p (n- 1).

Accordingly, we can derive the third term from the first two terms: f(n) = f(n- 1) * p+f(n-2) * q, which is called generalized Fibonacci sequence.

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

When p= 1 and q=2, we get the number of Pell-Pythagoras strings (the set of series related to a right triangle with an integer side length).

When p=- 1 and q=2, we get arithmetic progression. When f 1= 1 and f2=2, we get that the natural sequence 1, 2, 3, 4 ... is characterized by the difference between the square of each number and the product of the two numbers before and after it is 1 (the difference of arithmetic progression is called natural feature).

Fibonacci sequence p = 1 in a broad sense has similar golden characteristics, pythagorean characteristics and natural characteristics.

When f 1= 1, f2=2, p=2 and q= 1, we get the geometric series 1, 2,4,8, 16. ...

Related mathematics

permutation and combination

There is a flight of stairs with 10 steps, and it is stipulated that each step can only span one or two steps. How many different ways are there to climb 10 steps?

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

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

Similarly, a unified coin was thrown 10 times. How many possible situations are there for head discontinuity?

The answer is (1/√ 5) * {[(1+√ 5)/2] (10+2)-(1-√ 5)/2) (10+2.

Find the general formula A (1) = 1, a (n+1) =1/a (n).

Through mathematical induction, we can get: a(n)=F(n+ 1)/F(n). Substitute the general term of Fibonacci sequence and simplify it to get the result.

Rabbit reproduction problem

Fibonacci series is also called "rabbit series" because mathematician Leonardo Fibonacci introduced it by taking rabbit breeding as an example.

Generally speaking, rabbits can reproduce two months after birth, and a pair of rabbits can give birth to a pair of rabbits every month. If all rabbits don't die, how many pairs of rabbits can you breed in a year?

We might as well take a pair of newborn rabbits to analyze:

In the first month, the rabbits were infertile, so they were still a couple.

Two months later, a pair of rabbits were born with two pairs of logarithms.

Three months later, the old rabbit gave birth to another pair. Because rabbits have no reproductive ability, a pair is three.

-

By analogy, the following table can be listed:

Number of past months

1

2

three

four

five

six

seven

eight

nine

10

1 1

12

Logarithm of offspring

1

1

1

2

three

five

eight

13

2 1

34

55

Eighty-nine

Logarithm of adult rabbit

1

1

2

three

five

eight

13

2 1

34

55

Eighty-nine

144

Population logarithm

1

1

2

three

five

eight

13

2 1

34

55

Eighty-nine

144

233

Logarithm of young rabbits = Logarithm of adult rabbits in last month

Logarithm of adult rabbits = logarithms of adult rabbits last month+logarithms of young rabbits last month.

Logarithm of population = Logarithm of adult rabbits this month+Logarithm of young rabbits this month.

It can be seen that the logarithm of young people, the logarithm of adults and the logarithm of population all constitute a series. This series has a very obvious feature, that is, the sum of the two adjacent items in front constitutes the latter item.

This series was written by Italian mathematician Fibonacci in. Besides the property of a(n+2)=an+a(n+ 1), the general formula of this series can also be proved as an = (1/√ 5) * {[(1+√ 5)/.

Sequence and matrix

Fibonacci sequence 1, 1, 2, 3, 5, 8, 13,. has the following definitions.

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

F( 1)= 1

F(2)= 1

For the following matrix multiplication

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

Female (male) 10 female (male-1)

Its operation is to multiply the matrix 1 1 on the right by the matrix F(n):

10 Fahrenheit (n- 1)

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

F(n)=F(n)

It can be seen that the multiplication of this matrix completely conforms to the definition of Fibonacci sequence.

Let the matrix A= 1 1 iterate n times, and we can get: f (n+1) = a (n) * f (1) = a (n) *1.

1 0 F(n) F(0) 0

This is the definition of matrix multiplication of Fibonacci sequence.

Another algorithm of matrix multiplication, A n (n is an even number) = A (n/2) * A (n/2), so that we can realize matrix multiplication with logarithmic complexity through the idea of dichotomy.

Therefore, the answer can be obtained recursively.

Another solution of sequence value;

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

Where [x] represents the integer closest to x.