???? ??? ? C#? ???? ???? ??? ??? ??? ? ?????. ??? 0, 1, 1, 2, 3, 5, 8… ???? ??? 0? 1?? ???? ?? ??? ?? ? ??? ????. ???? ??? 13^????? ????? ??? ???(Leonardo Pisano Bigollo)? ????? ???. ???? ??? ?? ????? ?????. ????? ??? ?? ?? ??, ? ? ??? ??? ??? ?? ???? ? ???????. ???? ??? ??? ?? ??? ????.

???? ?? ??

???? ??? ????? ??? ?? ? ??? ????. ??? ???? ??? 0, 1, 1, 2, 3, 5, 8, 13, 21??? ???… ? ?? ??? ??? 13? 21? ?? ?? ? ??? ?? ???. ??? ?? ??? 13???. +21=34.

???? ??? ???? ??? ??? ????

F(n)= F(n-1) +F(n-2)

F(n)? ? ???? F(n-1) +F(n-2)? ?? ?? ????.

??? ??? 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…

??? ??? F(n)= F(n-1) +F(n-2)

F(n)= 55+89

F(n)= 144

?? ??? 144???.

???? ??? ??? ??? ??

???? ??? ??? ???? ??? ? ????.

1. ??? ??

? ??? ???? ???? ?? ?? ?????.

??:

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
namespaceFibonacciDemo
{
classProgram
{
staticint Fibonacci(int n)
{
intfirstnumber = 0, secondnumber = 1, result = 0;
if (n == 0) return 0; //It will return the first number of the series
if (n == 1) return 1; // it will return ?the second number of the series
for (int i = 2; i<= n; i++)? // main processing starts from here
{
result = firstnumber + secondnumber;
firstnumber = secondnumber;
secondnumber = result;
}
return result;
}
staticvoid Main(string[] args)
{
Console.Write("Length of the Fibonacci Series: ");
int length = Convert.ToInt32(Console.ReadLine());
for(int i = 0; i< length; i++)
{
Console.Write("{0} ", Fibonacci(i));
}
Console.ReadKey();
}
}
}

2. ??? ??

? ??? ???? ? ?? ?????.

?? 1

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
namespaceFibonacciDemo
{
classProgram
{
staticint Fibonacci(int n)
{
intfirstnumber = 0, secondnumber = 1, result = 0;
if (n == 0) return 0; //it will return the first number of the series
if (n == 1) return 1; // it will return the second number of the series
return Fibonacci(n-1) + Fibonacci(n-2);
}
staticvoid Main(string[] args)
{
Console.Write("Length of the Fibonacci Series: ");
int length = Convert.ToInt32(Console.ReadLine());
for(int i = 0; i< length; i++)
{
Console.Write("{0} ", Fibonacci(i));
}
Console.ReadKey();
}
}
}

?? 2

using System.Collections.Generic;
using System.Linq;
using System.Text;
namespace FibonacciSeries
{
class Program
{
public static void Fibonacci
(
int firstnumber,
int secondnumber,
int count,
int length,
)
{
if (count <= length)
{
Console.Write("{0} ", firstnumber);
Fibonacci(secondnumber, firstnumber + secondnumber, count + 1, length);
}
}
public static void Main(string[] args)
{
Console.Write("Length of the Fibonacci Series: ");
int length = Convert.ToInt32(Console.ReadLine());
Fibonacci(0, 1, 1, length);
Console.ReadKey();
}
}
}

??:

3. ??? ??? ????

??:

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
public class Program
{
public static int[] Fibonacci(int number)
{
int[] a = new int[number];
a[0] = 0;
a[1] = 1;
for (int i = 2; i < number; i++)
{
a[i] = a[i - 2] + a[i - 1];
}
return a;
}
public static void Main(string[] args)
{
var b = Fibonacci(10);
foreach (var elements in b)
{
Console.WriteLine(elements);
}
}
}

??:

???? ??? N?? ?? ?? ??? ??????

??? ??? ????

?? 1

??:

using System;
namespace FibonacciSeries
{
class Program {
public static int NthTerm(int n)
{
if ((n == 0) || (n == 1))
{
return n;
}
else
{
return (NthTerm(n - 1) + NthTerm(n - 2));
}
}
public static void Main(string[] args)
{
Console.Write("Enter the nth term of the Fibonacci Series: ");
int number = Convert.ToInt32(Console.ReadLine());
number = number - 1;
Console.Write(NthTerm(number));
Console.ReadKey();
}
}
}

? ??? ???? ??? n?? ?? ?? ?????. ?? ??, ????? 12^?? ??? ???? ??? 89? ???.

?? 2

(O(Log t) ??).

t?? ???? ?? ?? ? ??? ? ?? ? ?? ?? ??? ????. t? ???? = t/2:

F(t) = [2*F(k-1) + F(k)]*F(k)

t? ???? k = (t + 1)/2

F(t) = F(k)*F(k) + F(k-1)*F(k-1)

???? ??

??? ?? ??? (-1)t = Ft+1Ft-1 – Ft2? ????.

FmFt + Fm-1Ft-1 = Fm+t-1

t = t+1??? ??

FmFt+1 + Fm-1Ft = Fm+t

m = t

F2t-1 = Ft2 + Ft-12

F2t = (Ft-1 + Ft+1)Ft = (2Ft-1 + Ft)Ft

??? ?? ?? ??? ?????

t? ???? k = t/2? ?????

t? ???? k = (t+1)/2? ?????

??? ??? ???? STACK? ??? ??? ????? ???? ?? ??? ? ????. ?? O(n)? ?? ???? ?????. ?? ????? ???? ?????.

??:

int f(n) :
if( n==0 || n==1 )
return n;
else
return f(n-1) + f(n-2)

?? ?? ????? n=4? ?? ??? ?

fn(4)

f(3)???????????? f(2)

f(2)?? f(1)???? f(1)?? f(0)

f(1)?f(0)

?? ????. f(4)? ????? f(3)? f(2) ?? ???? ???. 4?? ?? ?? ?? f(2)? ? ? ???? f(1)? ? ? ?????. ? ?? ?? ?? ???? ??? ????.

f(n)? ????? ??? ??? ??? f(n+1)-1??? ??? ????.

??

??? ?? ??? ??? ??? ??? ???? ? ? ?? ?? ??? ????? ?? ?????. ???? ?? ??? ?? ??(? ?? ??)? ???? ??? ? ??? ? ?? ??? ???? ?? ?? ??? ???? ???? ?? ?????.

? ??? C#? ???? ??? ?? ?????. ??? ??? PHP ??? ????? ?? ?? ??? ?????!