Fibonacci numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4141, 6765, etc.
Lucas Number:
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5781, 9349, etc.
Pell Number:
0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, etc.
Pell - Lucas Number:
2, 2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, etc.
These are all well-known sequences in mathematics.
Fibonacci sequence
Introduction to the Fibonacci Sequence
The Fibonacci sequence (also translated as "Fibonacci sequence" or "Fibonacci sequence") is a very beautiful and harmonious sequence whose shape can be illustrated by a series of squares arranged in a spiral (such as right entry diagram), the initial square (shown in gray in the figure) has a side length of 1, and the side length of the square on its left is also 1, and another square is placed on top of these two squares, and its side length It is 2, and then add squares with side lengths of 3, 5, 8, 13, 21... and so on. Each of these numbers is equal to the sum of the previous two numbers, and they just form the Fibonacci sequence. The inventor of the "Fibonacci sequence" was the Italian mathematician Leonardo Fibonacci (born in 1170 AD and died in 1240. His native place was probably Pisa). He was known as the "Leonardo of Pisa". In 1202, he wrote the book "Liber Abaci" (Liber Abaci). He was the first European to study Indian and Arabic mathematical theories. His father was hired as a diplomatic consul by a business group in Pisa, where he was stationed in the 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.
The Fibonacci sequence refers to such a sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34...
The sequence begins with the third term, and each term is equal to the sum of the previous two. Its general term formula is: (1/√5)*{[(1 √5)/2]^n - [(1-√5)/2]^n} (√5 means the arithmetic square root of 5) ( 19th century French mathematician Minnie (Jacques Phillipe Marie Binet 1786-1856)
It is very interesting that such a sequence of natural numbers, the general term formula is actually expressed by irrational numbers.
The famous Fibonacci sequence may also be related to the American suspense writer Dan Brown, who cleverly used the sequence in his novel "The Da Vinci Code".
In fact, Yang Hui's triangle is mentioned in the current high school textbooks in our country, and the Fibonacci sequence can be found in it.
⋙ The emergence of the Fibonacci sequence
At the beginning of the 13th century, the best mathematician in Europe was Fibonacci; he wrote a book called "Book of Abacus", which was the best mathematics book in Europe at that time. There are many interesting math problems in the book, the most interesting of which is the following one:
"If a pair of rabbits can give birth to one pair of young rabbits every month, and each pair of young rabbits can start to give birth to another pair of young rabbits in the third month after its birth, assuming that there is no death, there will be one pair of young rabbits. Starting from a newborn rabbit, how many pairs of rabbits can be bred after one year?"
Fibonacci arranged the first few calculated numbers in a string: 1, 1, 2, 3, 5, 8...
There is a rule implied in this series of numbers: starting from the third number, each subsequent number is the sum of the two previous numbers. According to this law, as long as some simple additions are made, the number of rabbits in each month can be calculated.
Therefore, the numbers calculated according to this law constitute a famous sequence in the history of mathematics. Everyone calls it the "Fibonacci sequence", also known as the "rabbit sequence". This sequence has many peculiar properties. For example, starting from the third number, the ratio of each number to the number after it is very close to 0.618, which coincides with the famous "golden section law". People have also discovered that even the growth laws of some organisms can be described by this sequence under certain assumptions.
Fischer himself did not discuss this sequence further. It was not until the beginning of the 19th century that people studied it in detail. Around 1960, many mathematicians were very interested in the Fibonacci sequence and related phenomena. They not only established the Fibonacci Society, but also established related publications. Articles also multiply like Fibonacci's rabbits.
⋙ The origin and relationship of the Fibonacci sequence
The Fibonacci sequence comes from the rabbit problem, and it has a recursive relationship,
f(1)=1
f(2)=1
f(n)=f(n-1) f(n-2), where n>=2
{f(n)} is the Fibonacci sequence.
⋙ Fibonacci Sequence Formula
Its general term formula is: {[(1+√5)/2]^n - [(1－√5)/2]^n }/√5 (Note: √5 means root number 5)
⋙ Some properties of the Fibonacci sequence
1), f(n)f(n)-f(n 1)f(n-1)=(-1)^n;
2), f(1) f(2) f(3) ... f(n)=f(n 2)-1
3), arctan[1/f(2n 1)]=arctan[1/f(2n 2)] arctan[1/f(2n 3)]
【Existence of Fibonacci sequence】
It can even be said that the Fibonacci sequence is everywhere. Here are just a few common examples:
The sum of the numbers on the diagonal of Yang Hui's triangle constitutes the Fibonacci sequence.
Domino cards (which can be seen as a 2×1 square) completely cover an n×2 board, and the number of covered schemes is equal to the Fibonacci sequence.
From the perspective of bee reproduction, the andragon has only a mother and no father, because the eggs laid by the queen bee, the fertilized ones hatch into female bees, and the unfertilized ones hatch into androgens. When people trace the ancestors of Xiongfeng, they find that the number of the nth generation ancestors of a Xiongfeng is just the nth term Fn of the Fibonacci sequence.
The arrangement of the 13 chromatic scales of the piano is completely similar to that of the sixth generation of Xiongfeng, indicating that the tone is also related to the Fibonacci sequence.
The number of petals of some flowers in nature conforms to the Fibonacci sequence, that is to say, in most cases, the number of petals of a flower is 3, 5, 8, 13, 21, 34,... (there are 6 There are two sets of 3 pieces; 4 pieces may be genetic mutations).
If a branch grows a new branch every year, and the new branch grows a new branch every year after two years, then the number of branches over the years also constitutes a Fibonacci sequence.
【Fibonacci Sequence and Golden Section】
What is the relationship between the Fibonacci sequence and the golden section? It is found through research that the ratio of two adjacent Fibonacci numbers gradually tends to the golden ratio as the serial number increases. That is, f(n- 1)/f(n)-→0.618.... Since Fibonacci numbers are all integers, the quotient of dividing two integers is a rational number, so it is only gradually approaching the irrational number of the golden ratio. But when we continue to calculate the larger Fibonacci numbers later, we will find that the ratio of two adjacent numbers is indeed very close to the golden ratio.
Not only the "Fibonacci numbers" starting from 1, 1, 2, 3, 5... are like this, choose two integers at random, and then arrange them according to the rules of Fibonacci numbers, the ratio between the two numbers It will also gradually approach the golden ratio.
Triangle of the Padua Sequence
[Variant of the Fibonacci sequence]
1. Padua sequence: 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, ... Such a sequence is called the Padua sequence. It is very similar to the Fibonacci sequence, except that each number is obtained by skipping the number before it and adding the two numbers before it. This series of numbers can be represented by another picture, which is composed of some equilateral triangles (as shown on the right). The first triangle is shown in gray. In order to make these triangles fit together seamlessly, the first three triangles have side lengths of 1, the next two triangles have side lengths of 2, and then 3, 4, 5, and 7 in sequence. , 9, 12, 16, 2l...etc.
2. Dongdong has 15 pieces of candy. If he eats at least 3 pieces a day until he runs out, how many different ways to eat it?
If Dongdong has 3 candies, 4 candies or 5 candies, there is only one way to eat them; if there are 6 candies, there are 2 ways to eat them; if there are 7 candies, there are 3 ways to eat them; if If there are 8 candies, there are 4 ways to eat them; if there are 9 pieces of candies, there are 6 ways to eat them.
That is: the number of candy grains: 3 4 5 6 7 8 9 10 11 12. ．．
How to eat sugar: 1 1 1 2 3 4 6 9 13 19. ．．
Such a sequence is different from the Fibonacci sequence in that it skips the number in the middle every time, and then adds the 1st and 3rd numbers together to equal the 4th number. Its laws are both similar and different from the Fibonacci sequence.
3. Xiao Ming wants to go up the stairs. He can go up one, two or three steps at a time. If the stairs have 10 steps, how many different ways can he walk?
Here we might as well study the rules: if the stairs have only one level, he has 1 way of walking; if the stairs have two steps, he has 2 ways of walking; if the stairs have three steps, he has 4 ways of walking; If there are five stairs, he has 7 ways to walk.
That is: the number of steps of the stairs: 1 2 3 4 5 6 7 8 . ．．
How to go up stairs: 1 2 4 7 13 24 44 81. ．．
The rule here is that starting from the fourth number here, each number is equal to the sum of the three numbers before it.
[This sequence has many wonderful properties]
For example: as the number of sequence items increases, the ratio of the previous item to the next item is closer to the golden section 0.6180339887... (the ratio of the latter item to the previous item is 1.6180339887... )
There is another property, starting from the second term, the square of each odd term is 1 more than the product of the preceding two terms, and the square of each even term is 1 less than the product of the preceding two terms.
If you see such a topic: Someone cuts an 8*8 square into four pieces and puts them together into a 5*13 rectangle, and asks you in surprise: Why is 64=65? In fact, this property of the Fibonacci sequence is used: 5, 8, and 13 are the three adjacent items in the sequence. In fact, the areas of the two blocks before and after are indeed different by 1, but there is a slender line in the figure behind. The slit is not easy for ordinary people to notice.
If you choose any two numbers as the starting point, such as 5, -2.4, and then add the two items together to form 5, -2.4, 2.6, 0.2, 2.8, 3, 5.8, 8.8, 14.6...etc., you It will be found that with the development of the sequence, the ratio of the two terms before and after is getting closer and closer to the golden section, and the difference between the square of a certain term and the product of the two terms before and after also alternately differs by a certain value.
The nth item of the Fibonacci sequence also represents the number of all subsets in the set {1,2,...,n} that do not contain adjacent positive integers.
【Alias of Fibonacci sequence】
The Fibonacci sequence was introduced by the mathematician Leonardo Fibonacci with the example of rabbit breeding, so it is also called the "rabbit sequence".
Generally speaking, rabbits have the ability to reproduce after two months of birth, and a pair of rabbits can give birth to a pair of young rabbits every month. If none of the rabbits died, how many pairs of rabbits could be bred in one year? Let's analyze a pair of newly born rabbits:
In the first month, the bunny has no reproductive ability, so it is still a pair;
Two months later, there are two pairs of rabbits born;
Three months later, the old rabbit gave birth to another pair, because the younger rabbits have not yet been able to reproduce, so there are three pairs in total;
－－－－－
By analogy, the following table can be listed:
Elapsed months: 0 1 2 3 4 5 6 7 8 9 10 11 12
Rabbit pairs: 1 1 2 3 5 8 13 21 34 55 89 144 233
The numbers 1, 1, 2, 3, 5, 8 in the table --- form a sequence of numbers. This number sequence is related to a very obvious feature, that is: the sum of the preceding two adjacent items constitutes the latter item.
This sequence was proposed by the Italian medieval mathematician Fibonacci in "Complete Book of Abacus". The general term formula of this series, in addition to having the property of a(n 2)=an a(n 1)/, can also prove that The general term formula is: an=1/√[(1+√5/2) n-(1-√5/2) n](n=1,2,3.....)
Lucas sequence
There is a great relationship between Lucas Sequence and Fibonacci Sequence.
First define the integers P and Q such that D = P2 - 4Q > 0,
Thus we get an equation x2 - Px Q = 0, whose roots are a, b,
The Lucas sequence is now defined as:
Un(P,Q) = (an - bn) / (ab) and Vn(P,Q) = an bn
Where n is a non-negative integer, U0(P,Q) = 0, U1(P,Q) = 1, V0(P,Q) = 2, V1(P,Q) = P, …
We have the following identities related to Lucas numbers:
Um n = UmVn - anbnUm-n , Vm n = VmVn - anbnVm-n
Um 1 = P*Um - Q*Um-1 、 Vm 1 = P*Vm - Q*Vm-1 (n = 1)
U2n = UnVn, V2n = Vn2 - Qn
U2n 1 = Un 1Vn - Qn, V2n 1 = Vn 1Vn - PQn
If (P,Q) = (1,-1), we have Un as the Fibonacci number,
That is, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4141, 6765, etc.
And Vn is Lucas Number,
That is, 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5781, 9349, etc.
If (P,Q) = (2,-1), we have Un as the Pell Number,
That is, 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, etc.
And Vn is Pell-Lucas Number (Pell-Lucas Number) (see another article "Pell Sequence" for details),
That is, 2, 2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, etc.
These are all well-known sequences in mathematics.
Properties of Lucas numbers
Lucas numbers (abbreviated as Ln) have many properties similar to Fibonacci numbers. Such as Ln = Ln-1 Ln-2, where the difference is that L1 = 1, L2 = 3.
So Lucas numbers are: 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...... (OEIS A000204), the square numbers are only 1 and 4, which is determined by Proven by John HE Cohn. And the prime numbers, that is, the Lucas prime numbers (Lucas Prime) are: 3, 7, 11, 29, 47, ...... . Among them, the largest probable prime number (Probable Prime) is known to be L574219, which has as many as 120,005 digits.
We have the following identities related to Lucas numbers:
Ln2 - Ln-1Ln 1 = 5 (-1)n
L12 L22 ...... Ln2 = LnLn 1 - 2
Lm n = (5FmFn LmLn) / 2 (where Fn is Fibonacci number)
Lm-n = (-1)n (LmLn - 5FmFn) / 2
Ln2 - 5Fn2 = 4 (-1)n
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