Golomb Sequence | Set 2

Given a number N. The task is to find the first N terms of the Golomb Sequence. Golomb sequence is a non-decreasing integer sequence where the n-th term is equal to the number of times n appears in the sequence.

Input: N = 11
Output: 1 2 2 3 3 4 4 4 5 5 5
Explanation:
The first term is 1. 1 appears only once.
The second term is 2. 2 appears two times.
The third term is 2. 3 appears two times.
The fourth term is 3. 4 appears three times.
the fifth term is 3. 5 appears three times.

Input: N = 6
Output: 1 2 2 3 3 4

Approach:

  1. Initialise the array arr[] with [1] as the Golomb sequence starts with 1.
  2. Store the value of last index in map M.
  3. For Golomb sequence till N:
    • Intialise cnt = 0 and map the .
    • If cnt is not equals to 0 then, then current element in Golomb sequence is the previous element in the sequence and decrease the cnt.
    • Else the current element in Golomb sequence is equals to 1 + previous element in the sequence and cnt is updated as the value of map at current element in Golomb sequence and decrease the cnt.
    • Map the current index with the current value at Golomb Sequence.
  4. Print all the element of Golomb Sequence stored in array arr[].

Below is the implementation of the above approach:

CPP



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// C++ program to find the first
// N terms of Golomb Sequence
#include "bits/stdc++.h"
#define MAX 100001
using namespace std;
  
// Function to print the Golomb
// Sequence
void printGolombSequence(int N)
{
  
    // Initialise the array
    int arr[MAX];
  
    // Initialise the cnt to 0
    int cnt = 0;
  
    // First and second element
    // of Golomb Sequence is 0, 1
    arr[0] = 0;
    arr[1] = 1;
  
    // Map to store the count of
    // current element in Golomb
    // Sequence
    map<int, int> M;
  
    // Store the count of 2
    M[2] = 2;
  
    // Iterate over 2 to N
    for (int i = 2; i <= N; i++) {
  
        // If cnt is equals to 0
        // then we have new number
        // for Golomb Sequence
        // which is 1 + previous
        // element
        if (cnt == 0) {
            arr[i] = 1 + arr[i - 1];
            cnt = M[arr[i]];
            cnt--;
        }
  
        // Else the current element
        // is the previous element
        // in this Sequence
        else {
            arr[i] = arr[i - 1];
            cnt--;
        }
  
        // Map the current index to
        // current value in arr[]
        M[i] = arr[i];
    }
  
    // Print the Golomb Sequence
    for (int i = 1; i <= N; i++) {
        cout << arr[i] << ' ';
    }
}
  
// Driver Code
int main()
{
    int N = 11;
    printGolombSequence(N);
    return 0;
}

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Java

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// Java program to find the first
// N terms of Golomb Sequence
import java.util.*;
  
class GFG{
  
static int MAX = 1000;
  
// Function to print the Golomb
// Sequence
static void printGolombSequence(int N)
{
  
    // Initialise the array
    int []arr = new int[MAX];
    for(int i = 0; i < MAX; i++)
    arr[i] = 0;
      
    // Initialise the cnt to 0
    int cnt = 0;
  
    // First and second element
    // of Golomb Sequence is 0, 1
    arr[0] = 0;
    arr[1] = 1;
  
    // Map to store the count of
    // current element in Golomb
    // Sequence
    Map<Integer,Integer> M=new HashMap<Integer,Integer>(); 
      
    // Store the count of 2
    M.put(2,2);
  
    // Iterate over 2 to N
    for (int i = 2; i <= N; i++) {
  
        // If cnt is equals to 0
        // then we have new number
        // for Golomb Sequence 1 2 2 3 3 4 4 4 5 5 5
        // which is 1 + previous
        // element
        if (cnt == 0) {
            arr[i] = 1 + arr[i - 1];
            cnt = M.get(arr[i]);
            cnt--;
        }
  
        // Else the current element
        // is the previous element
        // in this Sequence
        else {
            arr[i] = arr[i - 1];
            cnt--;
        }
  
        // Map the current index to
        // current value in arr[]
            M.put(i, arr[i]);
    }
  
    // Print the Golomb Sequence
    for (int i = 1; i <= N; i++)
    {
        System.out.print(arr[i]+" ");
    }
}
  
// Driver Code
public static void main(String args[])
{
    int N = 11;
    printGolombSequence(N);
}
}
  
// This code is contributed by Surendra_Gangwar

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Python3

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# Python3 program to find the first
# N terms of Golomb Sequence
MAX = 100001
  
# Function to print the Golomb
# Sequence
def printGolombSequence(N):
  
    # Initialise the array
    arr = [0] * MAX
  
    # Initialise the cnt to 0
    cnt = 0
  
    # First and second element
    # of Golomb Sequence is 0, 1
    arr[0] = 0
    arr[1] = 1
  
    # Map to store the count of
    # current element in Golomb
    # Sequence
    M = dict()
  
    # Store the count of 2
    M[2] = 2
  
    # Iterate over 2 to N
    for i in range(2, N + 1):
  
        # If cnt is equals to 0
        # then we have new number
        # for Golomb Sequence
        # which is 1 + previous
        # element
        if (cnt == 0):
            arr[i] = 1 + arr[i - 1]
            cnt = M[arr[i]]
            cnt -= 1
  
        # Else the current element
        # is the previous element
        # in this Sequence
        else:
            arr[i] = arr[i - 1]
            cnt -= 1
  
        # Map the current index to
        # current value in arr[]
        M[i] = arr[i]
  
    # Print the Golomb Sequence
    for i in range(1, N + 1):
        print(arr[i], end=" ")
  
# Driver Code
N = 11
printGolombSequence(N)
  
# This code is contributed by mohit kumar 29

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Output:

1 2 2 3 3 4 4 4 5 5 5

Time Complexity: O(N)
Auxillary Space: O(N)

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