四面体数-yiteyi-C++库

一个数被称为四面体数，如果它可以表示为一个三角形底和三个边的金字塔，则称为四面体数。n ^th 四面体数是前n个数的和三角数 . 前十个四面体数是： 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, …

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图片[1]-四面体数-yiteyi-C++库

n的公式 ^th 四面体数：

T_n = (n * (n + 1) * (n + 2)) / 6

证据：

The proof uses the fact that the n^th tetrahedral number is given by,Tri_n = (n * (n + 1)) / 2It proceeds by induction.Base CaseT₁ = 1 = 1 * 2 * 3 / 6Inductive StepT_n+1 = T_n + Tri_n+1T_n+1 = [((n * (n + 1) * (n + 2)) / 6] + [((n + 1) * (n + 2)) / 2]T_n+1 = (n * (n + 1) * (n + 2)) / 6

以下是上述理念的实施：

C++

                         // CPP Program to find the                       
                         // nth tetrahedral number                       
                         #include <iostream>                       
                         using                                     namespace                                     std;                       
           
                         int                                     tetrahedralNumber(                                     int                                     n)                       
                         {                       
                                                 return                                     (n * (n + 1) * (n + 2)) / 6;                       
                         }                       
           
                         // Driver Code                       
                         int                                     main()                       
                         {                       
                                                 int                                     n = 5;                       
                                   
                                                 cout << tetrahedralNumber(n) << endl;                       
           
                                                 return                                     0;                       
                         }                       

JAVA

                         // Java Program to find the                       
                         // nth tetrahedral number                       
                         class                                     GFG {                       
                                   
                         // Function to find Tetrahedral Number                       
                         static                                     int                                     tetrahedralNumber(                                     int                                     n)                       
                         {                       
                                                 return                                     (n * (n +                                     1                                     ) * (n +                                     2                                     )) /                                     6                                     ;                       
                         }                       
           
                         // Driver Code                       
                         public                                     static                                     void                                     main(String[] args)                       
                         {                       
                                                 int                                     n =                                     5                                     ;                       
                                   
                                                 System.out.println(tetrahedralNumber(n));                       
                         }                       
                         }                       
           
                         // This code is contributed by Manish Kumar Rai.                       

python

                         # Python3 Program to find the                       
                         # nth tetrahedral number                       
           
                         def                                     tetrahedralNumber(n):                       
                                   
                                                 return                                     (n                                     *                                     (n                                     +                                     1                                     )                                     *                                     (n                                     +                                     2                                     ))                                     /                                     6                       
           
                         # Driver Code                       
                         n                                     =                                     5                       
                         print                                     (tetrahedralNumber(n))                       

C#

                         // C# Program to find the                       
                         // nth tetrahedral number                       
                         using                                     System;                       
           
                         public                                     class                                     GFG{                       
                                   
                                                 // Function to find Tetrahedral Number                       
                                                 static                                     int                                     tetrahedralNumber(                                     int                                     n)                       
                                                 {                       
                                                 return                                     (n * (n + 1) * (n + 2)) / 6;                       
                                                 }                       
                                   
                                                 // Driver code                       
                                                 static                                     public                                     void                                     Main ()                       
                                                 {                       
                                                 int                                     n = 5;                       
                                   
                                                 Console.WriteLine(tetrahedralNumber(n));                       
                                                 }                       
                         }                       
           
                         // This code is contributed by Ajit.                       

PHP

                         <?php                       
                         // PHP Program to find the                       
                         // nth tetrahedral number                       
           
                         function                                     tetrahedralNumber(                                     $n                                     )                       
                         {                       
                                                 return                                     (                                     $n                                     * (                                     $n                                     + 1) * (                                     $n                                     + 2)) / 6;                       
                         }                       
           
                         // Driver Code                       
                                                 $n                                     = 5;                       
           
                                                 echo                                     tetrahedralNumber(                                     $n                                     );                       
                                   
                         // This code is contributed by mits                       
                         ?>                       

Javascript

                         <script>                       
           
                         // JavaScript Program to find the                       
                         // nth tetrahedral number                       
           
                         // Function to find Tetrahedral Number                       
                         function                                     tetrahedralNumber(n)                       
                         {                       
                                                 return                                     (n * (n + 1) * (n + 2)) / 6;                       
                         }                       
                                   
                         // Driver code                       
                                                 let n = 5;                       
                                                 document.write(tetrahedralNumber(n));                       
                                   
                                                 // This code is contributed by code_hunt.                       
                         </script>                       

输出：

时间复杂性 ：O（1）。

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