Tutorialspoint.dev

Cassini’s Identity

Given a number N, the task is to evaluate below expression. Expected time complexity is O(1).

 f(n-1)*f(n+1) - f(n)*f(n)

Where f(n) is the n-th Fibonacci number with n >= 1. First few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, ………..i.e. (considering 0 as 0th Fibonacci number)

Examples :

Input : n = 5
Output : -1
f(5-1=4) = 3
f(5+1=6) = 8
f(5)*f(5)= 5*5 = 25
f(4)*f(6)- f(5)*f(5)= 24-25= -1


Although the task is simple i.e. find n-1th, nth and (n+1)-th Fibonacci numbers. Evaluate the expression and display the result. But this can be done in O(1) time using Cassini’s Identity which states that:



           f(n-1)*f(n+1) - f(n*n) = (-1)^n 

So, we don’t need to calculate any Fibonacci term,the only thing is to check whether n is even or odd.

How does above formula work?
The formula is based on matrix representation of Fibonacci numbers.
fibo

C/C++

// C++ implementation to demonstrate working
// of Cassini’s Identity 
#include<bits/stdc++.h>
using namespace std;
  
// Returns (-1)^n
int cassini(int n)
{
   return (n & 1) ? -1 : 1;
  
// Driver program
int main()
{  
   int n = 5;
   cout << cassini(n);
   return 0;

Java

// Java implementation to demonstrate working
// of Cassini’s Identity 
  
class Gfg
{
    // Returns (-1)^n
    static int cassini(int n)
    {
       return (n & 1) != 0 ? -1 : 1;
    
  
    // Driver method
    public static void main(String args[])
    {
         int n = 5;
         System.out.println(cassini(n));
    }
}

Python3

# Python implementation
# to demonstrate working
# of Cassini’s Identity 
  
# Returns (-1)^n
def cassini(n):
  
   return -1 if (n & 1) else 1
   
# Driver program
   
n = 5
print(cassini(n))
     
# This code is contributed
# by Anant Agarwal.

C#

// C# implementation to demonstrate 
// working of Cassini’s Identity
using System;
  
class GFG {
  
    // Returns (-1) ^ n
    static int cassini(int n)
    {
       return (n & 1) != 0 ? -1 : 1;
    
   
    // Driver Code
    public static void Main()
    {
         int n = 5;
         Console.Write(cassini(n));
    }
}
  
// This code is contributed by Nitin Mittal.

PHP

<?php
// PHP implementation to 
// demonstrate working of 
// Cassini’s Identity 
  
// Returns (-1)^n
function cassini($n)
{
    return ($n & 1) ? -1 : 1;
  
// Driver Code
$n = 5;
echo(cassini($n));
  
// This code is contributed by Ajit.
?>


Output :

-1

Reference :
https://en.wikipedia.org/wiki/Cassini_and_Catalan_identities

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.



This article is attributed to GeeksforGeeks.org

You Might Also Like

leave a comment

code

0 Comments

load comments

Subscribe to Our Newsletter