Moser-de Bruijn 数列

给定一个整数“n”，打印 Moser-de Bruijn 数列的前“n”项。
Moser-de Bruijn 序列是通过将数字 4(例如，1、4、16、64 等)的不同幂相加而获得的序列。

例子 ：

Input : 5
Output : 0 1 4 5 16

Input : 10
Output : 0 1 4 5 16 17 20 21 64 65

观察到序列的项遵循递推关系：

1) S(2 * n) = 4 * S(n)
2) S(2 * n + 1) = 4 * S(n) + 1
with S(0) = 0 and S(1) = 1

这里可以注意到，任何为 4 的非可分幂之和的数字都不是序列的一部分。例如，8 不是序列的一部分，因为它是由 4 的非可分幂(4 和 4)的总和形成的。
因此，任何不是 4 的幂并且出现在序列中的数字都必须是 4 的不同幂的总和。
例如，21 是序列的一部分，即使它不是 4 的幂，因为它是 4 的不同幂(1、4 和 16)的总和。

使用上面讨论的递归关系来有效地生成序列。

C++

// CPP code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
#include 
using namespace std;
 
// Function to generate nth term
// of Moser-de Bruijn Sequence
int gen(int n)
{
    // S(0) = 0
    if (n == 0)
        return 0;
     
    // S(1) = 1
    else if (n == 1)
        return 1;
     
    // S(2 * n) = 4 * S(n)
    else if (n % 2 == 0)
        return 4 * gen(n / 2);
     
    // S(2 * n + 1) = 4 * S(n) + 1
    else if (n % 2 == 1)
        return 4 * gen(n / 2) + 1;
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
void moserDeBruijn(int n)
{
    for (int i = 0; i < n; i++)
        cout << gen(i) << " ";
    cout << "\n";
}
 
// Driver Code
int main()
{
    int n = 15;
    cout << "First " << n << " terms of "
         << "Moser-de Bruijn Sequence : \n";
    moserDeBruijn(n);
    return 0;
}

Java

// Java code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
 
class GFG
{
     
// Function to generate nth term
// of Moser-de Bruijn Sequence
public static int gen(int n)
{
     
    // S(0) = 0
    if (n == 0)
        return 0;
     
    // S(1) = 1
    else if (n == 1)
        return 1;
     
    // S(2 * n) = 4 * S(n)
    else if (n % 2 == 0)
        return 4 * gen(n / 2);
     
    // S(2 * n + 1) = 4 * S(n) + 1
    else if (n % 2 == 1)
        return 4 * gen(n / 2) + 1;
    return 0;
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
public static void moserDeBruijn(int n)
{
    for (int i = 0; i < n; i++)
        System.out.print(gen(i) + " ");
    System.out.println();
}
 
// Driver Code
public static void main(String args[])
{
    int n = 15;
    System.out.println("First " + n +
                       " terms of " +
      "Moser-de Bruijn Sequence : ");
    moserDeBruijn(n);
}
}
 
// This code is contributed by JaideepPyne.

Python3

# Python code to generate first
# 'n' terms of the Moser-de
# Bruijn Sequence
 
# Function to generate nth term
# of Moser-de Bruijn Sequence
def gen(n):
 
    # S(0) = 0
    if n == 0:
        return 0
 
    # S(1) = 1
    elif n ==1:
        return 1
 
    # S(2 * n) = 4 * S(n)
    elif n % 2 ==0:
        return 4 * gen(n // 2)
 
    # S(2 * n + 1) = 4 * S(n) + 1
    elif n % 2 == 1:
        return 4 * gen(n // 2) +1
 
# Generating the first 'n' terms
# of Moser-de Bruijn Sequence
def moserDeBruijn(n):
    for i in range(n):
        print(gen(i), end = " ")
 
# Driver Program
n = 15
print("First", n, "terms of ",
       "Moser-de Brujn Sequence:")
moserDeBruijn(n)
 
# This code is contributed by Shrikant13

C#

// C# code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
using System;
 
class GFG {
     
// Function to generate nth term
// of Moser-de Bruijn Sequence
public static int gen(int n)
{
     
    // S(0) = 0
    if (n == 0)
        return 0;
     
    // S(1) = 1
    else if (n == 1)
        return 1;
     
    // S(2 * n) = 4 * S(n)
    else if (n % 2 == 0)
        return 4 * gen(n / 2);
     
    // S(2 * n + 1) = 4 * S(n) + 1
    else if (n % 2 == 1)
        return 4 * gen(n / 2) + 1;
    return 0;
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
public static void moserDeBruijn(int n)
{
    for (int i = 0; i < n; i++)
        Console.Write(gen(i) + " ");
        Console.WriteLine();
}
 
// Driver Code
public static void Main()
{
    int n = 15;
    Console.WriteLine("First " + n +
                    " terms of " +
    "Moser-de Bruijn Sequence : ");
    moserDeBruijn(n);
}
}
 
// This code is contributed by anuj_67.

Javascript

 
// Javascript code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
 
// Function to generate nth term
// of Moser-de Bruijn Sequence
function gen(n)
{
 
    // S(0) = 0
    if (n == 0)
        return 0;
 
    // S(1) = 1
    else if (n == 1)
        return 1;
 
    // S(2 * n) = 4 * S(n)
    else if (n % 2 == 0)
        return 4 * gen(parseInt(n / 2, 10));
 
    // S(2 * n + 1) = 4 * S(n) + 1
    else if (n % 2 == 1)
        return 4 * gen(parseInt(n / 2, 10)) + 1;
         
    return 0;
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
function moserDeBruijn(n)
{
    for(let i = 0; i < n; i++)
        document.write(gen(i) + " ");
         
    document.write("</br>");
}
 
// Driver code
let n = 15;
document.write("First " + n + " terms of " +
               "Moser-de Bruijn Sequence : " + "</br>");
moserDeBruijn(n);
 
// This code is contributed by decode2207
 

输出 ：

First 15 terms of Moser-de Bruijn Sequence : 
0 1 4 5 16 17 20 21 64 65 68 69 80 81 84

动态规划实现：

C++

// CPP code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
#include 
using namespace std;
 
// Function to generate nth term
// of Moser-de Bruijn Sequence
int gen(int n)
{
    int S[n+1];
 
    S[0] = 0;
    S[1] = 1;
 
    for (int i = 2; i <= n; i++)
    {   
        // S(2 * n) = 4 * S(n)
        if (i % 2 == 0)
           S[i] = 4 * S[i / 2];
     
        // S(2 * n + 1) = 4 * S(n) + 1
        else
           S[i] = 4 * S[i / 2] + 1;
    }
     
    return S[n];
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
void moserDeBruijn(int n)
{
    for (int i = 0; i < n; i++)
        cout << gen(i) << " ";
    cout << "\n";
}
 
// Driver Code
int main()
{
    int n = 15;
    cout << "First " << n << " terms of "
         << "Moser-de Bruijn Sequence : \n";
    moserDeBruijn(n);
    return 0;
}

Java

// Java code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
 
class GFG
{
     
// Function to generate nth term
// of Moser-de Bruijn Sequence
static int gen(int n)
{
    int []S = new int [n + 1];
 
    S[0] = 0;
    if(n != 0)
        S[1] = 1;
 
    for (int i = 2; i <= n; i++)
    {
         
        // S(2 * n) = 4 * S(n)
        if (i % 2 == 0)
        S[i] = 4 * S[i / 2];
     
        // S(2 * n + 1) = 4 * S(n) + 1
        else
        S[i] = 4 * S[i/2] + 1;
    }
     
    return S[n];
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
static void moserDeBruijn(int n)
{
    for (int i = 0; i < n; i++)
        System.out.print(gen(i)+" ");
}
 
// Driver Code
public static void main(String[] args)
{
    int n = 15;
    System.out.println("First " + n +
                       " terms of " +
      "Moser-de Bruijn Sequence : ");
    moserDeBruijn(n);
}
}
 
// This code is contributed by
// Smitha Dinesh Semwal.

蟒蛇3

# python3 code to generate first 'n' terms
# of the Moser-de Bruijn Sequence
 
# Function to generate nth term
# of Moser-de Bruijn Sequence
def gen( n ):
    S = [0, 1]
    for i in range(2, n+1):
         
        # S(2 * n) = 4 * S(n)
        if i % 2 == 0:
            S.append(4 * S[int(i / 2)]);
             
        # S(2 * n + 1) = 4 * S(n) + 1
        else:
            S.append(4 * S[int(i / 2)] + 1);
    z = S[n];
    return z;
 
# Generating the first 'n' terms
# of Moser-de Bruijn Sequence
def moserDeBruijn(n):
    for i in range(n):
        print(gen(i), end = " ")
 
# Driver Code
n = 15
print("First", n, "terms of ",
    "Moser-de Brujn Sequence:")
moserDeBruijn(n)
 
# This code is contributed by mits.

C＃

// C# code to generate first 'n' terms
// of the Moser-de Bruijn Sequence
using System;
 
class GFG
{
     
// Function to generate nth term
// of Moser-de Bruijn Sequence
static int gen(int n)
{
    int []S = new int [n + 1];
 
    S[0] = 0;
    if(n != 0)
        S[1] = 1;
 
    for (int i = 2; i <= n; i++)
    {
         
        // S(2 * n) = 4 * S(n)
        if (i % 2 == 0)
        S[i] = 4 * S[i / 2];
     
        // S(2 * n + 1) = 4 * S(n) + 1
        else
        S[i] = 4 * S[i/2] + 1;
    }
     
    return S[n];
}
 
// Generating the first 'n' terms
// of Moser-de Bruijn Sequence
static void moserDeBruijn(int n)
{
    for (int i = 0; i < n; i++)
        Console.Write(gen(i)+" ");
}
 
// Driver Code
public static void Main()
{
    int n = 15;
    Console.WriteLine("First " + n +
                      " terms of " +
     "Moser-de Bruijn Sequence : ");
    moserDeBruijn(n);
}
}
 
// This code is contributed by
// Smitha Dinesh Semwal.

Javascript

    // Javascript code to generate first 'n' terms
    // of the Moser-de Bruijn Sequence
     
    // Function to generate nth term
    // of Moser-de Bruijn Sequence
    function gen(n)
    {
        let S = new Array(n + 1);
        S.fill(0);
 
        S[0] = 0;
        if(n != 0)
            S[1] = 1;
 
        for (let i = 2; i <= n; i++)
        {
 
            // S(2 * n) = 4 * S(n)
            if (i % 2 == 0)
                S[i] = 4 * S[parseInt(i / 2, 10)];
 
            // S(2 * n + 1) = 4 * S(n) + 1
            else
                S[i] = 4 * S[parseInt(i / 2, 10)] + 1;
        }
 
        return S[n];
    }
 
    // Generating the first 'n' terms
    // of Moser-de Bruijn Sequence
    function moserDeBruijn(n)
    {
        for (let i = 0; i < n; i++)
            document.write(gen(i)+" ");
    }
     
    let n = 15;
    document.write("First " + n +
                      " terms of " +
     "Moser-de Bruijn Sequence : " + "</br>");
    moserDeBruijn(n);
 

输出 ：

First 15 terms of Moser-de Bruijn Sequence : 
0 1 4 5 16 17 20 21 64 65 68 69 80 81 84