Interesting facts about Fibonacci numbers

We know Fibonacci number, Fn = Fn-1 + Fn-2.

First few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …. .

Here are some interesting facts about Fibonacci number :

1. Pattern in Last digits of Fibonacci numbers :
Last digits of first few Fibonacci Numbers are :

0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, ... 

The series of last digits repeats with a cycle length of 60 (Refer this for explanations of this result).



C

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// C program to demonstrate that sequence of last 
// digits of Fibonacci numbers repeats after 60.
#include<stdio.h>
#define max 100
int main()
{
    long long int arr[max];
    arr[0] = 0;
    arr[1] = 1;
  
    // storing Fibonacci numbers
    for (int i = 2; i < max; i++)
        arr[i] = arr[i-1] + arr[i-2];
  
    // Traversing through store numbers
    for (int i = 1; i < max - 1; i++)
    {
        // Since first two number are 0 and 1
        // so, if any two consecutive number encounter 0 and 1
        // at their unit place, then it clearly means that
        // number is repeating/ since we just have to find
        // the sum of previous two number
        if ((arr[i] % 10 == 0) && (arr[i+1] % 10 == 1))
            break;
    }
    printf("Sequence is repeating after index %d", i);
}

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Java

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// Java program to demonstrate that sequence of last 
// digits of Fibonacci numbers repeats after 60. 
  
class GFG{
static int max=100
public static void main(String[] args) 
    long[] arr=new long[max]; 
    arr[0] = 0
    arr[1] = 1;
    int i=0;
  
    // storing Fibonacci numbers 
    for (i = 2; i < max; i++) 
        arr[i] = arr[i-1] + arr[i-2]; 
  
    // Traversing through store numbers 
    for (i = 1; i < max - 1; i++) 
    
        // Since first two number are 0 and 1 
        // so, if any two consecutive number encounter 0 and 1 
        // at their unit place, then it clearly means that 
        // number is repeating/ since we just have to find 
        // the sum of previous two number 
        if ((arr[i] % 10 == 0) && (arr[i+1] % 10 == 1)) 
            break
    
    System.out.println("Sequence is repeating after index "+i); 
}
// This code is conributed by mits

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Python3

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# Python3 program to demonstrate that sequence of last
# digits of Fibonacci numbers repeats after 60.
  
  
if __name__=='__main__':
    max = 100
    arr = [0 for i in range(max)]
    arr[0] = 0
    arr[1] = 1
  
# storing Fibonacci numbers
    for i in range(2, max):
        arr[i] = arr[i - 1] + arr[i - 2]
  
    # Traversing through store numbers
    for i in range(1, max - 1):
          
  
    # Since first two number are 0 and 1
    # so, if any two consecutive number encounter 0 and 1
    # at their unit place, then it clearly means that
    # number is repeating/ since we just have to find
    # the sum of previous two number
        if((arr[i] % 10 == 0) and (arr[i + 1] % 10 == 1)):
            break
  
    print("Sequence is repeating after index", i)
  
# This code is contributed by
# Sanjit_Prasad

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C#

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// C# program to demonstrate that sequence of last 
// digits of Fibonacci numbers repeats after 60. 
  
class GFG{
static int max=100; 
public static void Main() 
    long[] arr=new long[max]; 
    arr[0] = 0; 
    arr[1] = 1;
    int i=0;
  
    // storing Fibonacci numbers 
    for (i = 2; i < max; i++) 
        arr[i] = arr[i-1] + arr[i-2]; 
  
    // Traversing through store numbers 
    for (i = 1; i < max - 1; i++) 
    
        // Since first two number are 0 and 1 
        // so, if any two consecutive number encounter 0 and 1 
        // at their unit place, then it clearly means that 
        // number is repeating/ since we just have to find 
        // the sum of previous two number 
        if ((arr[i] % 10 == 0) && (arr[i+1] % 10 == 1)) 
            break
    
    System.Console.WriteLine("Sequence is repeating after index "+i); 
}
// This code is conributed by mits

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PHP

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<?php
// php program to demonstrate that
// sequence of last digits of 
// Fibonacci numbers repeats after
// 60. global $MAX=100
  
    $arr[0] = 0;
    $arr[1] = 1;
  
    // storing Fibonacci numbers
    for ($i = 2; $i < 100; $i++)
        $arr[$i] = $arr[$i-1] +
                       $arr[$i-2];
  
    // Traversing through store
    // numbers
    for ($i = 1; $i <100 - 1; $i++)
    {
        // Since first two number are
        // 0 and 1 so, if any two 
        // consecutive number encounter
        // 0 and 1 at their unit place,
        // then it clearly means that
        // number is repeating/ since 
        // we just have to find the
        // sum of previous two number
        if (($arr[$i] % 10 == 0) && 
                ($arr[$i+1] % 10 == 1))
            break;
    }
    echo "Sequence is repeating after",
                         " index ", $i;
  
// This code is contributed by ajit
?>

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