Here is a brief introduction.
Fibonacci numbers often appear in front of our eyes-for example, the arrangement of pinecones, pineapples and leaves, the number of petals of some flowers (typically sunflower petals), honeycombs, dragonfly wings, transcendental number E (more can be introduced), golden rectangle, golden section, equiangular spiral and twelve-average law.
With the increase of the number of items in the series, the ratio of the former item to the latter item approaches the golden section value of 0.6 18039887. ...
Starting from the second term, the square of each odd term is more than the product of the first two terms 1, and the square of each even term is n+Fn=Fn, fn-fn = f [0, 1] n = f [1] (n-65438
n 123456789 10…
fn 1459 1423376097 157…
fn 1347 1 1 18294776 123…
fn-fn 0 1 12358 132 134…
fn+fn 279 16254 166 107 173280…
(2) Any Fibonacci-Lucas sequence can be obtained by the sum of the finite terms of Fibonacci sequence, such as
n 123456789 10…
F[ 1， 1](n) 1 12358 132 13455…
F[ 1， 1](n- 1)0 1 12358 132 134…
F[ 1， 1](n- 1)0 1 12358 132 134…
fn 1347 1 1 18294776 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).
The generalized Fibonacci sequence p = 1 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. ...
Edit the related mathematics in this paragraph 1. Arrange and combine.
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.
2. The limit of the previous item compared with the latter item of two adjacent items in the sequence.
What is the limit of F(n)/F(n+ 1) when n tends to infinity?
This can be directly obtained from its general formula, and the limit is (-1+√5)/2, which is the numerical value of the golden section and a number representing natural harmony.
3. Find the general formula A (1) = 1, and 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.
3. The problem of rabbit reproduction (on the alias of Fibonacci sequence)
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 rabbits don't die, how many pairs of rabbits can you breed 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, there were two pairs of rabbits.
Three months later, the old rabbit gave birth to another pair. Because rabbits have no reproductive ability, a pair is three.
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By analogy, the following table can be listed:
The month has passed 0123456789101kloc-0/112.
Logarithm of offspring101123581321345589.
The logarithm of adult rabbit is 01123581321345 589144.
Logarithm of population1123581321345 589144233.
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 medieval mathematician Fibonacci in 186 1 year; ; In addition to the property that a(n+2)=an+a(n+ 1), the general formula of this sequence can also be proved as an = (1/√ 5) * {[(1+√ 5)/2.
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