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

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Improved By : nitin mittal, jit_t



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