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.  

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 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));
    }
}

# 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# 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 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.
?>

<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