Moser-de Bruijn Sequence
Given an integer ‘n’, print the first ‘n’ terms of the Moser-de Bruijn Sequence.
The Moser-de Bruijn sequence is the sequence obtained by adding up the distinct powers of the number 4 (For example 1, 4, 16, 64, etc).
Examples :
Input : 5 Output : 0 1 4 5 16 Input : 10 Output : 0 1 4 5 16 17 20 21 64 65
It is observed that the terms of the sequence follow the recurrence relation :
1) S(2 * n) = 4 * S(n) 2) S(2 * n + 1) = 4 * S(n) + 1 with S(0) = 0 and S(1) = 1
It may be noted here that any number which is the sum of non-distinct powers of 4 are not a part of the sequence. For example, 8 is not a part of the sequence because it is formed as the sum of non-distinct powers of 4, that are 4 and 4.
Thus, any number which is not a power of 4 and is present in the sequence must be the sum of the distinct powers of 4.
For example, 21 is a part of the sequence, even though it is not a power of 4 because it is the sum of the distinct powers of 4, that are 1, 4 and 16.
Employ the recurrence relation discussed above to generate the sequence, efficiently.
C++
// CPP code to generate first 'n' terms // of the Moser-de Bruijn Sequence #include <bits/stdc++.h> 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; } |
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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. |
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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 = 15print("First", n, "terms of ", "Moser-de Brujn Sequence:") moserDeBruijn(n) # This code is contributed by Shrikant13 |
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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. |
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PHP
<?php // PHP 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($n / 2); // S(2 * n + 1) = 4 * S(n) + 1 else if ($n % 2 == 1) return 4 * gen($n / 2) + 1; } // Generating the first 'n' terms // of Moser-de Bruijn Sequence function moserDeBruijn($n) { for ($i = 0; $i < $n; $i++) echo(gen($i) . " "); echo("\n"); } // Driver Code $n = 15; echo("First " . $n . " terms of " . "Moser-de Bruijn Sequence : \n"); echo(moserDeBruijn($n)); // This code is contributed by Ajit. ?> |
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Output :
First 15 terms of Moser-de Bruijn Sequence : 0 1 4 5 16 17 20 21 64 65 68 69 80 81 84
Dynamic Programming Implementation:
C++
// CPP code to generate first 'n' terms // of the Moser-de Bruijn Sequence #include <bits/stdc++.h> 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; } |
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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. |
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python3
# 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 = 15print("First", n, "terms of ", "Moser-de Brujn Sequence:") moserDeBruijn(n) # This code is contributed by mits. |
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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. |
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PHP
<?php // PHP 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 = array(); $S[0] = 0; $S[1] = 1; for ( $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 function moserDeBruijn( $n) { for ( $i = 0; $i < $n; $i++) echo gen($i) , " "; echo "\n"; } // Driver Code $n = 15; echo "First " ,$n ," terms of " , "Moser-de Bruijn Sequence : \n"; moserDeBruijn($n); // This code is contributed by anuj_67. ?> |
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Output :
First 15 terms of Moser-de Bruijn Sequence : 0 1 4 5 16 17 20 21 64 65 68 69 80 81 84
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