# 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) ? -1 : 1; ` `} ` ` ` `// Driver program ` `int` `main() ` `{ ` ` ` `int` `n = 5; ` ` ` `cout << cassini(n); ` ` ` `return` `0; ` `} ` |

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## 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)); ` ` ` `} ` `} ` |

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## 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. ` |

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## 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. ` |

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## 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. ` `?> ` |

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**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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