Math in algorithmic contests (i): Fibonacci series

Many of the recent topics are related to the Fabonacci sequence, and as an information group Konjac Konjac I have recently talked with the math group Lee a great God (Orz), including some of the nature of the Fabonacci series, to make a summary here.

Resources:

"Combinatorial Mathematics (5th Edition)", "Specific Mathematics (2nd edition)"

Body:

The Fibonacci sequence is shaped like 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ... of the sequence. The recursive form is defined as:

Sequence f[n]=f[n-1]+f[n-2], wherein f[0]=0,f[1]=1.

There are, of course, such Fibonacci sequences, which are shaped like:

G[n]=g[n-1]+g[n-2], but which g[0]∈z,g[1]∈z the sequence.

Widely used in the production of life, so in the informatics competition role can not be underestimated, this is some of the common Fibonacci series application problems:

Rabbit reproductive Problems: Oh, the rabbit headache;

The problem of the full-paved dominoes can also be said to be a step up.

First, a small code to find the Fibonacci sequence of Nth:

1 intFibonacciintN)2 {3     intFh=0, ft=1, fs,temp;4     if(n==0)return 0;5     if(n==1)return 1; 6      for(intI=1; i<n;++i)7     {8fs=fh+ft;9Fh=FS;Tenft=fh; One     } A     returnFS; -}

Of course, recursive or recursive algorithms can also be used, the following gives the recursive method of the code for the calculation:

1 intFibonacciintN)2 {3  　　    if(n==0)return 0; 4  　　    if(n==1)return 1; 5  　　    return(Fibonacci (n1) +fibonacci (n2)); 6}

Here are some of the properties of some of the Fibonacci series that have recently been seen:

The first is the general formula:

And the derivation of it:

There is also an important property:

GCD (f (n), F (m)) =GCD (n,m);

This nature to use number theory to prove, but unfortunately this konjac Konjac has not learned number theory, can not personally give proof, but this site has a proof method, interested can go to see:

http://www.douban.com/group/topic/33566582/

So the generation of Fibonacci sequence is very simple;

Although using its general formula involves a large number of powers and irrational numbers, at least when n is large, high precision can be used to ensure that the algorithm complexity is Linear order O (n),

It's easier than recursive, recursive, and cyclic versions of the build anyway.

Then many problems can be solved.

Math in algorithmic contests (i): Fibonacci series