Who knows what a rabbit sequence is?

Who knows what a rabbit sequence is?

[Rabbit series]
　　
The inventor of Fibonacci series is Italian mathematician Leonardo Fibonacci, who was born in 1170 and died in 1240. His native place is probably Pisa. He is known as "Leonardo of Pisa"
Fibonacci sequence refers to such a sequence: 0,1,1,2,3,5,8,13,21
This sequence starts from the third term, and each term is equal to the sum of the first two terms
(1 / √ 5) * {[(1 + √ 5) / 2] ^ n - [(1 - √ 5) / 2] ^ n} [√ 5 means root 5]
It's interesting that the general formula of such a series of numbers is expressed by irrational numbers
[the sequence has many wonderful properties]
For example, as the number of items in a sequence increases, the ratio of the former to the latter approaches the golden section of 0.6180339887
Another property is that, starting from the second term, the square of every odd term is one more than the product of the preceding and the following two terms, and the square of every even term is one less than the product of the preceding and the following two terms
If you see a question like this: someone cut an 8 * 8 square into four pieces and put it together into a 5 * 13 rectangle, pretending to be surprised to ask you: why 64 = 65? In fact, it uses the property of Fibonacci sequence: 5, 8 and 13 are the three adjacent terms in the sequence. In fact, the area difference between the front and the back is 1, but there is a slender slit in the back picture, It's not easy for ordinary people to notice
If you pick any two numbers as the starting point, such as 5, - 2.4, and then add the two terms to form 5, - 2.4, 2.6, 0.2, 2.8, 3, 5.8, 8.8, 14.6 You will find that with the development of the sequence, the ratio of the two terms is closer to the golden section, and the difference between the square of a term and the product of the two terms also alternates by a certain value
The nth term of Fibonacci sequence also represents the number of all subsets in the set {1,2,..., n} that do not contain adjacent positive integers
[Fibonacci sequence alias]
Fibonacci sequence was introduced by mathematician Leonardo Fibonacci with rabbit breeding as an example, so it is also called "rabbit sequence"
Fibonacci sequence
Generally speaking, two months after birth, rabbits have the ability to reproduce. A pair of rabbits can produce a pair of rabbits every month. If all rabbits do not die, how many pairs of rabbits can be bred in one year?
Let's take a pair of new born rabbits and analyze them
In the first month, the rabbits had no reproductive capacity, so they were still a pair;
Two months later, two pairs of rabbits were born;
Three months later, the old rabbit gave birth to another pair, because the little rabbit had no reproductive capacity, so there were three pairs;
　　－－－－－－
The following table can be listed by analogy:
Number of months: 0 12 34 5 6 7 8 9 10 11 12
Rabbit logarithm: 1 1 2 35 8 13 21 34 55 89 144 233
The numbers 0,1,1,2,3,5,8 - in the table form a sequence. This sequence has a very obvious feature: the sum of the preceding two adjacent terms forms the latter
This series was put forward by Fibonacci, an Italian mathematician in the middle ages, in the abacus book. The general term formula of this series not only has the property of a (n + 2) = an + a (n + 1 /), but also can prove that the general term formula is: an = 1 / √ [(1 + √ 5 / 2) n - (1 - √ 5 / 2) n] (n = 1,2,3.)
[derivation of general term formula of Fibonacci sequence]
Fibonacci sequence: 0,1,1,2,3,5,8,13,21
Let f (n) be the nth term of the sequence (n ∈ n +)
　　F(1)=F(2)=1,F(n)=F(n-1)+F(n-2)(n≥3)
Obviously, this is a linear recurrence sequence
Derivation method of general term formula 1: using characteristic equation
The characteristic equation of linear recurrence sequence is as follows
　　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
The solution is C1 = 1 / √ 5, C2 = - 1 / √ 5
Ψ f (n) = (1 / √ 5) * {[(1 + √ 5) / 2] ^ n - [(1 - √ 5) / 2] ^ n} [√ 5 denotes radical 5]
General formula derivation method 2: general method
Let R, s be constant
Let f (n) - R * f (n-1) = s * [f (n-1) - R * f (n-2)]
Then R + S = 1, - rs = 1
When n ≥ 3, there is
　　F(n)-r*F(n-1)=s*[F(n-1)-r*F(n-2)]
　　F(n-1)-r*F(n-2)=s*[F(n-2)-r*F(n-3)]
　　F(n-2)-r*F(n-3)=s*[F(n-3)-r*F(n-4)]
　　……
　　F(3)-r*F(2)=s*[F(2)-r*F(1)]
Multiply the above n-2 formulas to obtain:
　　F(n)-r*F(n-1)=[s^(n-2)]*[F(2)-r*F(1)]
　　∵s=1-r,F(1)=F(2)=1
The above formula can be simplified as follows:
　　F(n)=s^(n-1)+r*F(n-1)
Then:
　　F(n)=s^(n-1)+r*F(n-1)
　　= s^(n-1) +r*s^(n-2) + r^2*F(n-2)
　　= s^(n-1) +r*s^(n-2) + r^2*s^(n-3) + r^3*F(n-3)
　　……
　　= s^(n-1) +r*s^(n-2) + r^2*s^(n-3) +…… + r^(n-2)*s + r^(n-1)*F(1)
　　= s^(n-1) +r*s^(n-2) + r^2*s^(n-3) +…… + r^(n-2)*s + r^(n-1)
(this is the sum of the items of an equal ratio sequence with S ^ (n-1) as the first term, R ^ (n-1) as the last term and R / s as the tolerance.)
　　=[s^(n-1)-r^(n-1)*r/s]/(1-r/s)
　　=(s^n - r^n)/(s-r)
A solution of R + S = 1, - rs = 1 is s = (1 + √ 5) / 2, r = (1 - √ 5) / 2
Then f (n) = (1 / √ 5) * {[(1 + √ 5) / 2] ^ n - [(1 - √ 5) / 2] ^ n}
　　
Fibonacci sequence (f (n), f (0) = 0, f (1) = 1, f (2) = 1, f (3) = 2...) Other properties of the project:
　　1.f(0)+f(1)+f(2)+… +f(n)=f(n+2)-1
　　2.f(1)+f(3)+f(5)+… +f(2n-1)=f(2n)-1
　　3.f(0)+f(2)+f(4)+… +f(2n)=f(2n+1)-1
　　4.[f(0)]^2+[f(1)]^2+… +[f(n)]^2=f(n)·f(n+1)
　　5.f(0)-f(1)+f(2)-… +(-1)^n·f(n)=(-1)^n·[f(n+1)-f(n)]+1
　　6.f(m+n)=f(m-1)·f(n-1)+f(m)·f(n)
　　7.[f(n)]^2=(-1)^(n-1)+f(n-1)·f(n+1)
　　8.f(2n-1)=[f(n)]^2-[f(n-2)]^2
Fibonacci sequence is hidden in Yanghui triangle
　　1
　　1 1
　　1 2 1
　　1 3 3 1
　　1 4 6 4 1
　　……
After passing the "1" in the first line, make a 45 degree diagonal line to the left and down, and then make a parallel line of the straight line. Add up the number of each straight line to get a sequence of 1, 1, 2, 3, 5, 8
　　
(1) The number of petals of the following flowers has Fibonacci Number: Trillium, rose, Sanguinaria, cosmos, Goldilocks, columbine, Lily and iris
(2) the similar petals of the following flowers also have Fibonacci Numbers: Ziwan, Daphne, daisy
The Fibonacci number is often combined with the number of petals
　　3……………………… Lily and iris
　　5……………………… Columbine, Goldilocks, swallowwort
　　8……………………… Delphinium
　　13……………………… Calendula officinalis
　　21……………………… Fleabane
34,55,84…………… Daisies
(3) Fibonacci number can also be found in the arrangement of leaves, branches and stems
For example, select a leaf on a branch of a tree, record it as 0, and then count the leaves in order (assuming there is no damage) until it reaches the position directly opposite to that leaf, In a cycle, the ratio of the number of leaves to the number of turns of a leaf is called the ratio of leaf order (derived from Hera, meaning the arrangement of leaves). Most of the leaf order ratios are presented as the ratio of Fibonacci number
(4) Fibonacci sequence and gold ratio
Sequence of ratios of successive Fibonacci Numbers:
The limit of the sequence is. This connection implies that wherever the golden ratio, golden rectangle or equiangular spiral appears (especially in natural phenomena), the Fibonacci number will also appear, and vice versa
But each term is an integer. And the ratio of the two adjacent terms in the sequence is closer to 0.618. This sequence has a wide range of applications, such as the number of annual branches of trees follows the law of Fibonacci sequence. Moreover, the development of computer science provides a new application place for Fibonacci sequence

Why is Fibonacci sequence called rabbit sequence

Fibonacci series is the beginning of rabbit breeding as an example. Generally speaking, rabbits have reproductive capacity after two months of birth. A pair of rabbits can give birth to a pair of rabbits every month. If all rabbits do not die, then how many pairs of rabbits can be bred after a year? We might as well take a pair of new born rabbits to analyze: first

How many rabbits can be born in a year?

This cross Fibonacci sequence
It's just adding up one by one
The rule is as follows
1,1,2,3,5,8,13,21,34,55,89,144,233
But because of how often rabbits are born, we don't know exactly which one to take
Once a month,
The twelfth number is 144
A: 144 rabbits will be born in one year

Number sequence 1.1.2.3.5.8.13 ··········································································
Can you make it clear? I'm still a primary school student

You can divide several known numbers by three to get the remainder of 1,1,2,0,2,2,1,0
It can be observed that the remainder is a series of numbers in this cycle
So the number 2001 divided by three is one