Cassini’s Identity

3

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
Image Source : https://en.wikipedia.org/wiki/Cassini_and_Catalan_identities

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));
	}
}


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.

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