# Alcuin’s Sequence

Alcuin sequence is the expansion of

This series has significant importance as

• Alcuin Sequence a(n) is the number of triangles with integer sides and the perimeter of the triangle is n.
• Alcuin Sequence a(n) is the number of triangles with distinct integer sides and the perimeter of the triangle is n+6.

The Alcuin sequence is as follows:

0, 0, 1, 0, 1, 1, 2, 1, 3, 2

Examples:

Input: n = 10
Output: 0, 0, 1, 0, 1,
1, 2, 1, 3, 2
Input: n = 15
Output:0, 0, 1, 0, 1,
1, 2, 1, 3, 2,
4, 3, 5, 4, 7,


Approach:

• Find the nth term of the Alcuin sequence using
a(n) = round(n^2/12) – floor(n/4)*floor((n+2)/4)
• Find out the ith term and display it.

Below is the implementation of the above approach:

## C++

 #include  using namespace std;     // find the nth term  of  // Alcuin's sequence  int Alcuin(int n)  {      double _n = n, ans;         ans = round((_n * _n) / 12)            - floor(_n / 4)                  * floor((_n + 2) / 4);         // return the ans      return ans;  }     // print first n terms of Alcuin number  void solve(int n)  {      int i = 0;         for (int i = 1; i <= n; i++) {             // display the number          cout << Alcuin(i) << ", ";      }  }     // Driver code  int main()  {      int n = 15;      solve(n);      return 0;  }

## Java

 // Java program for Alcuin's Sequence  import java.util.*;     class GFG   {     // find the nth term of  // Alcuin's sequence  static int Alcuin(int n)  {      double _n = n, ans;         ans = Math.round((_n * _n) / 12) -             Math.floor(_n / 4) *             Math.floor((_n + 2) / 4);         // return the ans      return (int) ans;  }     // print first n terms of Alcuin number  static void solve(int n)  {      int i = 0;         for (i = 1; i <= n; i++)       {             // display the number          System.out.print(Alcuin(i) + ", ");      }  }     // Driver code  public static void main(String[] args)   {      int n = 15;      solve(n);  }  }     // This code is contributed by Princi Singh

## Python3

 # Python3 program for Alcuin’s Sequence  from math import ceil, floor     # find the nth term of  # Alcuin's sequence  def Alcuin(n):         _n = n      ans = 0        ans = (round((_n * _n) / 12) -              floor(_n / 4) *              floor((_n + 2) / 4))         # return the ans      return ans     # print first n terms of Alcuin number  def solve(n):         for i in range(1, n + 1):             # display the number          print(Alcuin(i), end = ", ")         # Driver code  n = 15 solve(n)     # This code is contributed by Mohit Kumar

## C#

 // C# program for Alcuin's Sequence  using System;         class GFG   {         // find the nth term of      // Alcuin's sequence      static int Alcuin(int n)      {          double _n = n, ans;             ans = Math.Round((_n * _n) / 12) -                 Math.Floor(_n / 4) *                 Math.Floor((_n + 2) / 4);             // return the ans          return (int) ans;      }         // print first n terms of Alcuin number      static void solve(int n)      {          int i = 0;             for (i = 1; i <= n; i++)           {                 // display the number              Console.Write(Alcuin(i) + ", ");          }      }         // Driver code      public static void Main(String[] args)       {          int n = 15;          solve(n);      }  }     // This code is contributed by Rajput-Ji

Output:

0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7,


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