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)"

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