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Print N terms of Withoff Sequence
  • Last Updated : 15 Feb, 2021

Wythoff array is an infinite matrix of integers derived from the Fibonacci sequence. Every positive integer in the matrix occurs only once. 
Wythoff array: 
 

  1    2    3    5    8    13    ...
  
  4    7    11   18   29   47    ...
                        
  6    10   16   26   42   68    ...
   
  9    15   24   39   63   102   ...
  
  12   20   32   52   84   136   ...
  
  14   23   37   60   97   157   ...

  .    .    .    .    .    .
  .    .    .    .    .    . 

If Am, n denotes the element in the mth row and nth column then 
 

  • Am, 1 = [[mφ]φ]
  • Am, 2 = [[mφ]φ2]
  • Am, n = Am, n-2 + Am, n-1 for n > 2
  • φ = (1 + √5) / 2

If we traverse matrix in an anti-diagonal way starting from top-left element then 
Wythoff sequence: 
 

1, 2, 4, 3, 7, 6, 5, 11, 10, 9…. 

For a given N, the task to print first N numbers of the sequence.
Examples: 
 



Input : N = 10 
Output : 1, 2, 4, 3, 7, 6, 5, 11, 10, 9 
Input : N = 15 
Output : 1, 2, 4, 3, 7, 6, 5, 11, 10, 9, 8, 18, 16, 15, 12 
 

 

Approach: 
The above recusrions can be modified as 
 

  • T(n, -1) = n-1, if k = -1
  • T(n, 0) = [n*φ], if k = 0
  • T(n, k) = T(n, k-1) + T(n, k-2), if k > 0
  • φ = (1 + √5) / 2

So we can recursively find the value of T(n, k) with two base cases for t = 0 and for t = –1. we will store the values in a map and use it when needed to reduce computation. After we get the array we have to traverse it in an anti- diagonal way, so we set i=0 and j=0 and decrease the j and increase i when the j < 0 we initialise j = i and i = 0
we also keep a count which is increased when a number is displayed. We break the array when the count reaches the required value.
Below is the implementation of the above approach :
 

CPP




// C++ program to find Wythoff array
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the n, k term of Wythoff array
int Wythoff(map<int, map<int, int> >& mp, int n, int k)
{
    // tau = (sqrt(5)+1)/2
    double tau = (sqrt(5) + 1) / 2.0, t_n_k;
 
    // Already_stored
    if (mp[n][k] != 0)
        return mp[n][k];
 
    // T(n, -1) = n-1.
    if (k == -1) {
        return n - 1;
    }
     
    // T(n, 0) = floor(n*tau).
    else if (k == 0) {
        t_n_k = floor(n * tau);
    }
     
    // T(n, k) = T(n, k-1) + T(n, k-2) for k>=1.
    else
    {
        t_n_k = Wythoff(mp, n, k - 1) +
                             Wythoff(mp, n, k - 2);
    }
 
    // Store
    mp[n][k] = t_n_k;
 
    // Return the ans
    return (int)t_n_k;
}
 
// Function to find  first n terms of Wythoff
// array by traversing in anti-diagonal
void Wythoff_Array(int n)
{
    int i = 0, j = 0, count = 0;
 
    // Map to store the Wythoff array
    map<int, map<int, int> > mp;
 
    while (count < n) {
 
        cout << Wythoff(mp, i + 1, j + 1);
        count++;
         
        if(count != n)
            cout << ", ";
 
        // Anti diagonal
        i++;
        j--;
 
        if (j < 0) {
            j = i;
            i = 0;
        }
    }
}
 
// Driver code
int main()
{
    int n = 15;
     
    // Function call
    Wythoff_Array(n);
     
    return 0;
}

Java




// Java program to find Wythoff array
import java.util.*;
public class GFG
{
 
  // Function to find the n, k term of Wythoff array
  static int Wythoff(HashMap<Integer,
                     HashMap<Integer, Integer>> mp,
                     int n, int k)
  {
 
    // tau = (sqrt(5)+1)/2
    double tau = (Math.sqrt(5) + 1) / 2.0, t_n_k;
 
    // Already_stored
    if (mp.containsKey(n) &&
        mp.get(n).containsKey(k) &&
        mp.get(n).get(k) != 0)
      return mp.get(n).get(k);
 
    // T(n, -1) = n-1.
    if (k == -1)
    {
      return n - 1;
    }
 
    // T(n, 0) = floor(n*tau).
    else if (k == 0)
    {
      t_n_k = Math.floor(n * tau);
    }
 
    // T(n, k) = T(n, k-1) + T(n, k-2) for k>=1.
    else
    {
      t_n_k = Wythoff(mp, n, k - 1) +
        Wythoff(mp, n, k - 2);
    }
 
    // Store
    mp.put(n, new HashMap<Integer, Integer>(k,(int)t_n_k));
 
    // Return the ans
    return (int)t_n_k;
  }
 
  // Function to find  first n terms of Wythoff
  // array by traversing in anti-diagonal
  static void Wythoff_Array(int n)
  {
    int i = 0, j = 0, count = 0;
 
    // Map to store the Wythoff array
    HashMap<Integer, HashMap<Integer,Integer>> mp =
      new HashMap<Integer, HashMap<Integer,Integer>>();
    while (count < n)
    {
      System.out.print(Wythoff(mp, i + 1, j + 1));
      count++;
      if(count != n)
        System.out.print(", ");
 
      // Anti diagonal
      i++;
      j--;
 
      if (j < 0)
      {
        j = i;
        i = 0;
      }
    }
  }
 
  // Driver code
  public static void main(String[] args)
  {
    int n = 15;
 
    // Function call
    Wythoff_Array(n);
  }
}
 
// This code is contributed by divyeshrabadiya07.

Python3




# Python3 program to find Wythoff array
import math
 
# Function to find the n, k term of Wythoff array
def Wythoff(mp, n, k):
 
    # tau = (sqrt(5)+1)/2
    tau = (math.sqrt(5) + 1) / 2
    t_n_k = 0
 
    # Already_stored
    if ((n in mp) and (k in mp[n])):
        return mp[n][k];
 
    # T(n, -1) = n-1.
    if (k == -1):
        return n - 1;
     
    # T(n, 0) = floor(n*tau).
    elif (k == 0):
        t_n_k = math.floor(n * tau);
     
    # T(n, k) = T(n, k-1) + T(n, k-2) for k>=1.
    else:
        t_n_k = Wythoff(mp, n, k - 1) + Wythoff(mp, n, k - 2)
     
    # Store
    if n not in mp:
        mp[n] = dict()
    mp[n][k] = t_n_k;
 
    # Return the ans
    return int(t_n_k)
 
# Function to find  first n terms of Wythoff
# array by traversing in anti-diagonal
def Wythoff_Array(n):
 
    i = 0
    j = 0
    count = 0;
 
    # Map to store the Wythoff array
    mp = dict()
 
    while (count < n):
 
        print(Wythoff(mp, i + 1, j + 1), end = '')
        count += 1
         
        if(count != n):
            print(", ", end = '')
 
        # Anti diagonal
        i += 1
        j -= 1
 
        if (j < 0):
            j = i;
            i = 0;
         
# Driver code
if __name__=='__main__':
 
    n = 15;
     
    # Function call
    Wythoff_Array(n);
 
    # This code is contributed by rutvik_56

C#




// C# program to find Wythoff array
using System;
using System.Collections.Generic;
class GFG
{
 
  // Function to find the n, k term of Wythoff array
  static int Wythoff(Dictionary<int, Dictionary<int, int>> mp, int n, int k)
  {
 
    // tau = (sqrt(5)+1)/2
    double tau = (Math.Sqrt(5) + 1) / 2.0, t_n_k;
 
    // Already_stored
    if (mp.ContainsKey(n) && mp[n].ContainsKey(k) && mp[n][k] != 0)
      return mp[n][k];
 
    // T(n, -1) = n-1.
    if (k == -1) {
      return n - 1;
    }
 
    // T(n, 0) = floor(n*tau).
    else if (k == 0)
    {
      t_n_k = Math.Floor(n * tau);
    }
 
    // T(n, k) = T(n, k-1) + T(n, k-2) for k>=1.
    else
    {
      t_n_k = Wythoff(mp, n, k - 1) + Wythoff(mp, n, k - 2);
    }
 
    // Store
    if(!mp.ContainsKey(n))
    {
      mp[n] = new Dictionary<int,int>();
    }
    mp[n][k] = (int)t_n_k;
 
    // Return the ans
    return (int)t_n_k;
  }
 
  // Function to find  first n terms of Wythoff
  // array by traversing in anti-diagonal
  static void Wythoff_Array(int n)
  {
    int i = 0, j = 0, count = 0;
 
    // Map to store the Wythoff array
    Dictionary<int, Dictionary<int,int>> mp = new Dictionary<int, Dictionary<int,int>>();
    while (count < n)
    {
 
      Console.Write(Wythoff(mp, i + 1, j + 1));
      count++;
      if(count != n)
        Console.Write(", ");
 
      // Anti diagonal
      i++;
      j--;
 
      if (j < 0) {
        j = i;
        i = 0;
      }
    }
  }
 
  // Driver code
  static void Main()
  {
    int n = 15;
 
    // Function call
    Wythoff_Array(n);
  }
}
 
// This code is contributed by divyesh072019.

Output: 
 

1, 2, 4, 3, 7, 6, 5, 11, 10, 9, 8, 18, 16, 15, 12, 

Reference : https://oeis.org/A035513
 

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