从斯特恩的双原子级数中找到第n个元素

给定整数n。我们必须找到斯特恩双原子级数的第n个项。
斯特恩的双原子序列是生成以下整数序列0、1、1、2、1、3、2、3、1、4、3、5、2、5、3、4，…的序列。它出现在Calkin-Wilf树中。有时也称为fusc函数。
用数学术语来说，斯特恩双原子序列的序列P(n)由递归关系定义。
$\\ p(n) = p(n/2) \hspace{5.5cm} for \ n \ is \ even\\ p(n) = p((n-1)/2)+p(n+1)/2) \hspace{2cm} for \ n \ is \ odd \\ \\ where \ p(0) = 0 \ and \ p(1) = 1 \\ \\ Stern's \ Diatomic \ Series \ 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, ......$
例子 ：

Input : n = 7
Output : 3

Input : n = 15
Output : 4

方法 ：
我们使用非常简单的动态规划概念来解决此问题，该概念用于查找斐波那契数。保存DP [0] = 0，DP [1] = 1的基本情况之后，我们必须简单地从i = 2遍历到n并根据斯特恩双原子级数的解释定义计算DP [i]。最后返回DP [n]的值。
算法 ：

// SET the Base case
    DP[0] = 0;
    DP[1] = 1;

    // Traversing the array from 2nd Element to nth Element
    for (int i=2; i<=n; i++)
    {
        // Case 1: for even n
        if (i%2 == 0)
            DP[i] = DP[i/2];

        // Case 2: for odd n
        else
            DP[i] = DP[(i-1)/2] + DP[(i+1)/2];
    }
    return DP[n];

C++

// Program to find the nth element
// of Stern's Diatomic Series
#include 
using namespace std;
 
// function to find nth stern'
// diatomic series
int findSDSFunc(int n)
{
    // Initializing the DP array
    int DP[n+1];
 
    // SET the Base case
    DP[0] = 0;
    DP[1] = 1;
 
    // Traversing the array from
    // 2nd Element to nth Element
    for (int i = 2; i <= n; i++) {
         
        // Case 1: for even n
        if (i % 2 == 0)
            DP[i] = DP[i / 2];
         
        // Case 2: for odd n
        else
            DP[i] = DP[(i - 1) / 2] +
                        DP[(i + 1) / 2];
    }
    return DP[n];
}
 
// Driver program
int main()
{
    int n = 15;   
    cout << findSDSFunc(n) << endl;   
    return 0;
}

Java

// Java program to find the nth element
// of Stern's Diatomic Series
 
class GFG {
     
    // function to find nth stern'
    // diatomic series
    static int findSDSFunc(int n)
    {
         
        // Initializing the DP array
        int DP[] = new int[n+1];
     
        // SET the Base case
        DP[0] = 0;
        DP[1] = 1;
     
        // Traversing the array from
        // 2nd Element to nth Element
        for (int i = 2; i <= n; i++)
        {
             
            // Case 1: for even n
            if (i % 2 == 0)
                DP[i] = DP[i / 2];
             
            // Case 2: for odd n
            else
                DP[i] = DP[(i - 1) / 2] +
                            DP[(i + 1) / 2];
        }
         
        return DP[n];
    }
     
    // Driver program
    public static void main(String[] args)
    {
        int n = 15;
         
        System.out.println(findSDSFunc(n));
    }
}
 
// This code is contributed by Smita Semwal.

Python 3

# Program to find the nth element
# of Stern's Diatomic Series
 
# function to find nth stern'
# diatomic series
def findSDSFunc(n):
 
    # Initializing the DP array
    DP = [0] * (n+1)
 
    # SET the Base case
    DP[0] = 0
    DP[1] = 1
 
    # Traversing the array from
    # 2nd Element to nth Element
    for i in range(2, n+1):
         
        # Case 1: for even n
        if (int(i % 2) == 0):
            DP[i] = DP[int(i / 2)]
         
        # Case 2: for odd n
        else:
            DP[i] = (DP[int((i - 1) / 2)]
                  + DP[int((i + 1) / 2)])
     
    return DP[n]
 
 
# Driver program
n = 15
 
print(findSDSFunc(n))
 
# This code is contribute by
# Smitha Dinesh Semwal

C#

// C# program to find the nth element
// of Stern's Diatomic Series
using System;
 
class GFG
{
    // function to find nth
    // stern' diatomic series
    static int findSDSFunc(int n)
    {
         
        // Initializing the DP array
        int []DP = new int[n + 1];
     
        // SET the Base case
        DP[0] = 0;
        DP[1] = 1;
     
        // Traversing the array from
        // 2nd Element to nth Element
        for (int i = 2; i <= n; i++)
        {
             
            // Case 1: for even n
            if (i % 2 == 0)
                DP[i] = DP[i / 2];
             
            // Case 2: for odd n
            else
                DP[i] = DP[(i - 1) / 2] +
                        DP[(i + 1) / 2];
        }
         
        return DP[n];
    }
     
    // Driver Code
    static public void Main ()
    {
        int n = 15;
        Console.WriteLine(findSDSFunc(n));
    }
}
 
// This code is contributed by aj_36

Javascript

 
// JavaScript program to find the nth element
// of Stern's Diatomic Series
 
    // function to find nth stern'
    // diatomic series
    function findSDSFunc(n)
    {
           
        // Initializing the DP array
        let DP = [];
       
        // SET the Base case
        DP[0] = 0;
        DP[1] = 1;
       
        // Traversing the array from
        // 2nd Element to nth Element
        for (let i = 2; i <= n; i++)
        {
               
            // Case 1: for even n
            if (i % 2 == 0)
                DP[i] = DP[i / 2];
               
            // Case 2: for odd n
            else
                DP[i] = DP[(i - 1) / 2] +
                            DP[(i + 1) / 2];
        }
        return DP[n];
    }
       
 
// Driver code
        let n = 15;     
        document.write(findSDSFunc(n));
           
          // This code is contributed by souravghosh0416.

输出：

4