Skip to content
Related Articles

Related Articles

Improve Article

Cassini’s Identity

  • Difficulty Level : Medium
  • Last Updated : 24 Mar, 2021

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)

Attention reader! Don’t stop learning now. Get hold of all the important mathematical concepts for competitive programming with the Essential Maths for CP Course at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

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.
?>

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

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

This article is contributed by Sahil Chhabra. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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




My Personal Notes arrow_drop_up
Recommended Articles
Page :