Cassini’s Identity - GeeksforGeeks

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)
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++
Java
Python3
C#
PHP
JavaScript
// C++ implementation to demonstrate working// of Cassini’s Identity #include<bits/stdc++.h>using namespace std;  // Returns (-1)^nint cassini(int n){   return (n & 1) ? -1 : 1;}   // Driver programint main(){     int n = 5;   cout << cassini(n);   return 0;} 
// 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)^ndef cassini(n):     return -1 if (n & 1) else 1   # Driver program   n = 5print(cassini(n))     # This code is contributed# by Anant Agarwal.
// C# implementation to demonstrate // working of Cassini’s Identityusing 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)^nfunction 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>
-1

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