Cassini’s Identity

Last Updated : 15 Sep, 2023

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.

C/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) != 0 ? -1 : 1; } 
  
// Driver Method 
int main() 
{ 
    int n = 5; 
    cout << (cassini(n)); 
  
    return 0; 
} 
  
// This code is contributed by phasing17

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. 
?> 

JavaScript

<script> 
// Javascript implementation to  
// demonstrate working of  
// Cassini’s Identity  
  
// Returns (-1)^n  
function cassini(n)  
{  
    return (n & 1) ? -1 : 1;  
}  
  
// Driver Code  
let n = 5;  
document.write(cassini(n));  
  
// This code is contributed by _saurabh_jaiswal. 
  
</script> 

Output :

-1

Time complexity: O(1) since only constant operations are performed

Auxiliary Space: O(1)

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

Suggest improvement

Fibonacci Search

Hosoya's Triangle

Share your thoughts in the comments

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Variations of Fibonacci number

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