# Telephone Number

In mathematics, the telephone numbers involution numbers are a sequence of integers are a sequence of integers that count the number of connection patterns in a telephone system with n subscribers, where connections are made between pairs of subscribers. These numbers also describe the number of matchings of a complete graph of n vertices, the number of permutations on n elements that are involutions, the sum of absolute value of coefficients of the Hermite polynomials, the number of standard Young tableaux with n cells, and the sum of the degrees of the irreducible representations of the symmetric group.

The telephone numbers are also used to count the number of ways to place n rooks on an n x n chessboard in such a way that no two rooks attack each other and in such a way the configuration of the rooks is symmetric under a diagonal reflection of the board.

The telephone number can be evaluated by the following recurrence relation:

Given a positive integer n. The task is to find the nth telephone number.
Examples :

```Input : n = 4
Output : 10

Input : n = 6
Output : 76
```

## Recommended: Please solve it on PRACTICE first, before moving on to the solution.

Below is naive implementation of finding the nth telephone number based on above recursive formula.

## C++

 `// CPP Program to find the nth telephone number. ` `#include ` `using` `namespace` `std; ` ` `  `// return the nth telephone number ` `int` `telephonenumber(``int` `n) ` `{ ` `    ``// base step ` `    ``if` `(n == 0 || n == 1) ` `        ``return` `1; ` ` `  `    ``// recursive step ` `    ``return` `telephonenumber(n - 1) +  ` `          ``(n - 1) * telephonenumber(n - 2); ` `} ` ` `  `// Driven Program ` `int` `main() ` `{ ` `    ``int` `n = 6; ` `    ``cout << telephonenumber(n) << endl; ` `    ``return` `0; ` `} `

## Java

 `// JAVA Code to find the nth ` `// telephone number. ` `import` `java.util.*; ` ` `  `class` `GFG { ` `     `  `    ``// return the nth telephone number ` `    ``static` `int` `telephonenumber(``int` `n) ` `    ``{ ` `        ``// base step ` `        ``if` `(n == ``0` `|| n == ``1``) ` `            ``return` `1``; ` `      `  `        ``// recursive step ` `        ``return` `telephonenumber(n - ``1``) +  ` `              ``(n - ``1``) * telephonenumber(n - ``2``); ` `    ``} ` `     `  `    ``/* Driver program to test above function */` `    ``public` `static` `void` `main(String[] args)  ` `    ``{ ` `        ``int` `n = ``6``; ` `        ``System.out.println(telephonenumber(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by Arnav Kr. Mandal.     `

## Python3

 `# Python3 code to find the  ` `# nth telephone number. ` ` `  `# return the nth telephone number ` `def` `telephonenumber (n): ` ` `  `    ``# base step ` `    ``if` `n ``=``=` `0` `or` `n ``=``=` `1``: ` `        ``return` `1` `         `  `    ``# recursive step ` `    ``return` `(telephonenumber(n ``-` `1``) ``+` `(n ``-` `1``) ` `            ``*` `telephonenumber(n ``-` `2``)) ` ` `  `# Driven Program ` `n ``=` `6` `print``(telephonenumber(n)) ` ` `  `# This code is contributed by "Sharad_Bhardwaj". `

## C#

 `// C# Code to find the nth ` `// telephone number. ` `using` `System; ` ` `  `class` `GFG { ` ` `  `    ``// return the nth telephone number ` `    ``static` `int` `telephonenumber(``int` `n) ` `    ``{ ` `        ``// base step ` `        ``if` `(n == 0 || n == 1) ` `            ``return` `1; ` ` `  `        ``// recursive step ` `        ``return` `telephonenumber(n - 1) +  ` `            ``(n - 1) * telephonenumber(n - 2); ` `    ``} ` ` `  `    ``/* Driver program to test above function */` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``int` `n = 6; ` `         `  `        ``Console.Write(telephonenumber(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by vt_m. `

## PHP

 ` `

Output :

```76
```

Below is efficient implementation of finding the nth telephone number using Dynamic Programming:

## C++

 `// CPP Program to find the nth telephone number. ` `#include ` `using` `namespace` `std; ` ` `  `// return the nth telephone number ` `int` `telephonenumber(``int` `n) ` `{ ` `    ``int` `dp[n + 1]; ` `    ``memset``(dp, 0, ``sizeof``(dp)); ` ` `  `    ``// Base case ` `    ``dp[0] = dp[1] = 1; ` ` `  `    ``// finding ith telephone number, where 2 <= i <= n. ` `    ``for` `(``int` `i = 2; i <= n; i++) ` `        ``dp[i] = dp[i - 1] + (i - 1) * dp[i - 2]; ` ` `  `    ``return` `dp[n]; ` `} ` ` `  `// Driver Program ` `int` `main() ` `{ ` `    ``int` `n = 6; ` `    ``cout << telephonenumber(n) << endl; ` `    ``return` `0; ` `} `

## Java

 `// JAVA Code to find nth Telephone Number ` `import` `java.util.*; ` ` `  `class` `GFG { ` `     `  `    ``// return the nth telephone number ` `    ``static` `int` `telephonenumber(``int` `n) ` `    ``{ ` `        ``int` `dp[] = ``new` `int``[n + ``1``]; ` `        `  `        ``// Base case ` `        ``dp[``0``] = dp[``1``] = ``1``; ` `      `  `        ``// finding ith telephone number, ` `        ``// where 2 <= i <= n. ` `        ``for` `(``int` `i = ``2``; i <= n; i++) ` `            ``dp[i] = dp[i - ``1``] + (i - ``1``) * dp[i - ``2``]; ` `      `  `        ``return` `dp[n]; ` `    ``} ` `     `  `    ``/* Driver program to test above function */` `    ``public` `static` `void` `main(String[] args)  ` `    ``{ ` `         ``int` `n = ``6``; ` `         ``System.out.println(telephonenumber(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by Arnav Kr. Mandal. `

## Python3

 `# Python3 code to find the ` `# nth telephone number. ` ` `  `# return the nth telephone number ` `def` `telephonenumber (n): ` `    ``dp ``=` `[``0``] ``*` `(n ``+` `1``) ` `     `  `    ``# Base case ` `    ``dp[``0``] ``=` `dp[``1``] ``=` `1` `     `  `    ``# finding ith telephone number, ` `    ``# where 2 <= i <= n. ` `    ``for` `i ``in` `range``(``2``, n ``+` `1``): ` `        ``dp[i] ``=` `dp[i ``-` `1``] ``+` `(i ``-` `1``) ``*` `dp[i ``-` `2``] ` `         `  `    ``return` `dp[n] ` `     `  `# Driver Code ` `n ``=` `6` `print``(telephonenumber(n)) ` ` `  `# This code is contributed by "Sharad_Bhardwaj". `

## C#

 `// C# Code to find nth Telephone Number ` `using` `System; ` ` `  `class` `GFG { ` ` `  `    ``// return the nth telephone number ` `    ``static` `int` `telephonenumber(``int` `n) ` `    ``{ ` `        ``int``[] dp = ``new` `int``[n + 1]; ` ` `  `        ``// Base case ` `        ``dp[0] = dp[1] = 1; ` ` `  `        ``// finding ith telephone number, ` `        ``// where 2 <= i <= n. ` `        ``for` `(``int` `i = 2; i <= n; i++) ` `            ``dp[i] = dp[i - 1] + (i - 1) * dp[i - 2]; ` ` `  `        ``return` `dp[n]; ` `    ``} ` ` `  `    ``/* Driver program to test above function */` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``int` `n = 6; ` `         `  `        ``Console.Write(telephonenumber(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by vt_m. `

## PHP

 ` `

Output :

```76
```

My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

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 Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Improved By : vt_m

Article Tags :
Practice Tags :

Be the First to upvote.

Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.