🏆 ยอดนิยม: web//https://www.geeksforgeeks.org/farey sequence

Farey sequence is a sequence which is generated for order n. The sequence has all rational numbers in range [0/0 to 1/1] sorted in increasing order such that the denominators are less than or equal to n and all numbers are in reduced forms i.e., 4/4 cannot be there as it can be reduced to 1/1.

Examples:

F1 = 0/1, 1/1
F2 = 0/1, 1/2, 1/1
F3 = 0/1, 1/3, 1/2, 2/3, 1/1
.
.
F7 = 0/1, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5,
     3/7, 1/2, 4/7, 3/5, 2/3, 5/7, 3/4, 4/5,
     5/6, 6/7, 1/1

Farey sequence is used in rational approximations of irrational numbers, ford circles and
in Riemann hypothesis (See this for more details)

How to generate Farey Sequence of given order?
The idea is simple, we consider every possible rational number from 1/1 to n/n. And for every generated rational number, we check if it is in reduced form. If yes, then we add it to Farey Sequence. A rational number is in reduced form if GCD of numerator and denominator is 1.

Below is the implementation based on above idea.

C++

// C++ program to print Farey Sequence of given order 
#include <bits/stdc++.h> 
using namespace std; 
  
// class for x/y (a term in farey sequence 
class Term { 
public: 
    int x, y; 
  
    // Constructor to initialize x and y in x/y 
    Term(int x, int y) 
        : x(x), y(y) 
    { 
    } 
}; 
  
// Comparison function for sorting 
bool cmp(Term a, Term b) 
{ 
    // Comparing two ratio 
    return a.x * b.y < b.x * a.y; 
} 
  
// GCD of a and b 
int gcd(int a, int b) 
{ 
    if (b == 0) 
        return a; 
    return gcd(b, a % b); 
} 
  
// Function to print Farey sequence of order n 
void farey(int n) 
{ 
    // Create a vector to store terms of output 
    vector<Term> v; 
  
    // One by one find and store all terms except 0/1 and n/n 
    // which are known 
    for (int i = 1; i <= n; ++i) { 
        for (int j = i + 1; j <= n; ++j) 
  
            // Checking whether i and j are in lowest term 
            if (gcd(i, j) == 1) 
                v.push_back(Term(i, j)); 
    } 
  
    // Sorting the term of sequence 
    sort(v.begin(), v.end(), cmp); 
  
    // Explicitly printing first term 
    cout << "0/1 "; 
  
    // Printing other terms 
    for (int i = 0; i < v.size(); ++i) 
        cout << v[i].x << "/" << v[i].y << " "; 
  
    // explicitely printing last term 
    cout << "1/1"; 
} 
  
// Driver program 
int main() 
{ 
    int n = 7; 
    cout << "Farey Sequence of order " << n << " is\n"; 
    farey(n); 
    return 0; 
} 

Python3

# Python3 program to print 
# Farey Sequence of given order 
  
# class for x/y (a term in farey sequence 
class Term: 
  
    # Constructor to initialize 
    # x and y in x/y 
    def __init__(self, x, y): 
        self.x = x 
        self.y = y 
  
# GCD of a and b 
def gcd(a, b): 
    if b == 0: 
        return a 
    return gcd(b, a % b) 
  
# Function to print  
# Farey sequence of order n 
def farey(n): 
  
    # Create a vector to  
    # store terms of output 
    v = [] 
  
    # One by one find and store  
    # all terms except 0/1 and n/n 
    # which are known 
    for i in range(1, n + 1): 
        for j in range(i + 1, n + 1): 
  
            # Checking whether i and j 
            # are in lowest term 
            if gcd(i, j) == 1: 
                v.append(Term(i, j)) 
  
    # Sorting the term of sequence 
    for i in range(len(v)): 
        for j in range(i + 1, len(v)): 
            if (v[i].x * v[j].y > v[j].x * v[i].y): 
                v[i], v[j] = v[j], v[i] 
  
    # Explicitly printing first term 
    print("0/1", end = " ") 
  
    # Printing other terms 
    for i in range(len(v)): 
        print("%d/%d" % (v[i].x,  
                        v[i].y), end = " ") 
  
    # explicitely printing last term 
    print("1/1") 
  
# Driver Code 
if __name__ == "__main__": 
    n = 7
    print("Farey sequence of order %d is" % n) 
    farey(n) 
  
# This code is contributed by 
# sanjeev2552 

Output:

Farey Sequence of order 7 is
0/1 1/7 1/6 1/5 1/4 2/7 1/3 2/5 3/7 1/2 4/7 
3/5 2/3 5/7 3/4 4/5 5/6 6/7 1/1

Time complexity of above approach is O(n² Log n) where O(log n) is an upper bound on time taken by Euclid’s algorithm for GCD.

Farey Sequence has below properties [See wiki for details]

A term x/y can be recursively evaluated using previous two terms. Below is the formula to compute x_n+2/y_n+2 from x_n+1/y_n+1 and x_n/y_n.

x[n+2] = floor((y[n]+n) / y[n+1])x[n+1]– x[n]      
y[n+2] = floor((y[n]+n) / y[n+1])y[n+1]– y[n]

We can use above properties to optimize.

C++

// Efficient C++ program to print Farey Sequence of order n 
#include <bits/stdc++.h> 
using namespace std; 
  
// Optimized function to print Farey sequence of order n 
void farey(int n) 
{ 
    // We know first two terms are 0/1 and 1/n 
    double x1 = 0, y1 = 1, x2 = 1, y2 = n; 
  
    printf("%.0f/%.0f %.0f/%.0f", x1, y1, x2, y2); 
  
    double x, y = 0; // For next terms to be evaluated 
    while (y != 1.0) { 
        // Using recurrence relation to find the next term 
        x = floor((y1 + n) / y2) * x2 - x1; 
        y = floor((y1 + n) / y2) * y2 - y1; 
  
        // Print next term 
        printf(" %.0f/%.0f", x, y); 
  
        // Update x1, y1, x2 and y2 for next iteration 
        x1 = x2, x2 = x, y1 = y2, y2 = y; 
    } 
} 
  
// Driver program 
int main() 
{ 
    int n = 7; 
    cout << "Farey Sequence of order " << n << " is\n"; 
    farey(n); 
    return 0; 
} 

Java

// Efficient Java program to print  
// Farey Sequence of order n 
class GFG 
{ 
  
// Optimized function to print  
// Farey sequence of order n 
static void farey(int n) 
{ 
    // We know first two terms are 0/1 and 1/n 
    double x1 = 0, y1 = 1, x2 = 1, y2 = n; 
  
    System.out.printf("%.0f/%.0f %.0f/%.0f", x1, y1, x2, y2); 
  
    double x, y = 0; // For next terms to be evaluated 
    while (y != 1.0)  
    { 
        // Using recurrence relation to find the next term 
        x = Math.floor((y1 + n) / y2) * x2 - x1; 
        y = Math.floor((y1 + n) / y2) * y2 - y1; 
  
        // Print next term 
        System.out.printf(" %.0f/%.0f", x, y); 
  
        // Update x1, y1, x2 and y2 for next iteration 
        x1 = x2; 
        x2 = x; 
        y1 = y2; 
        y2 = y; 
    } 
} 
  
// Driver program 
public static void main(String[] args) 
{ 
    int n = 7; 
    System.out.print("Farey Sequence of order " + n + " is\n"); 
    farey(n); 
} 
} 
  
// This code is contributed by Rajput-Ji 

Python3

# Efficient Python3 program to print 
# Farey Sequence of order n 
import math 
  
# Optimized function to print Farey  
# sequence of order n 
def farey(n): 
      
    # We know first two terms are  
    # 0/1 and 1/n 
    x1 = 0;  
    y1 = 1;  
    x2 = 1;  
    y2 = n; 
      
    print(x1, end = "")  
    print("/", end = "") 
    print(y1, x2, end = "")  
    print("/", end = "")  
    print(y2, end = " "); 
  
    # For next terms to be evaluated 
    x = 0; 
    y = 0;  
    while (y != 1.0): 
          
        # Using recurrence relation to 
        # find the next term 
        x = math.floor((y1 + n) / y2) * x2 - x1; 
        y = math.floor((y1 + n) / y2) * y2 - y1; 
  
        # Print next term 
        print(x, end = "") 
        print("/", end = "") 
        print(y, end = " "); 
  
        # Update x1, y1, x2 and y2 for  
        # next iteration 
        x1 = x2;  
        x2 = x;  
        y1 = y2;  
        y2 = y; 
  
# Driver Code 
n = 7; 
print("Farey Sequence of order", n, "is"); 
farey(n); 
  
# This code is contributed by mits 

PHP

<?php 
// Efficient php program to print 
// Farey Sequence of order n 
  
// Optimized function to print  
// Farey sequence of order n 
function farey($n) 
{ 
      
    // We know first two  
    // terms are 0/1 and 1/n 
    $x1 = 0;  
    $y1 = 1;  
    $x2 = 1;  
    $y2 = $n; 
      
        echo $x1, "/", $y1,  
             " ", $x2, "/",  
             $y2, " "; 
  
    // For next terms 
    // to be evaluated 
    $x; 
    $y = 0;  
    while ($y != 1.0) 
    { 
          
        // Using recurrence relation to 
        // find the next term 
        $x = floor(($y1 + $n) / $y2) * $x2 - $x1; 
        $y = floor(($y1 + $n) / $y2) * $y2 - $y1; 
  
        // Print next term 
        echo $x, "/", $y, " "; 
  
        // Update x1, y1, x2 and 
        // y2 for next iteration 
        $x1 = $x2;  
        $x2 = $x;  
        $y1 = $y2;  
        $y2 = $y; 
    } 
} 
  
    // Driver Code 
    $n = 7; 
    echo "Farey Sequence of order ", $n, " is\n"; 
    farey($n); 
  
// This code is contributed by ajit 
?> 

C#

// Efficient C# program to print  
// Farey Sequence of order n 
using System; 
  
public class GFG 
{ 
   
// Optimized function to print  
// Farey sequence of order n 
static void farey(int n) 
{ 
    // We know first two terms are 0/1 and 1/n 
    double x1 = 0, y1 = 1, x2 = 1, y2 = n; 
   
    Console.Write("{0:F0}/{1:F0} {2:F0}/{3:F0}", x1, y1, x2, y2); 
   
    double x, y = 0; // For next terms to be evaluated 
    while (y != 1.0)  
    { 
        // Using recurrence relation to find the next term 
        x = Math.Floor((y1 + n) / y2) * x2 - x1; 
        y = Math.Floor((y1 + n) / y2) * y2 - y1; 
   
        // Print next term 
        Console.Write(" {0:F0}/{1:F0}", x, y); 
   
        // Update x1, y1, x2 and y2 for next iteration 
        x1 = x2; 
        x2 = x; 
        y1 = y2; 
        y2 = y; 
    } 
} 
   
// Driver program 
public static void Main(String[] args) 
{ 
    int n = 7; 
    Console.Write("Farey Sequence of order " + n + " is\n"); 
    farey(n); 
} 
} 
  
// This code is contributed by 29AjayKumar 

Output:

Farey Sequence of order 7 is
0/1 1/7 1/6 1/5 1/4 2/7 1/3 2/5 3/7 1/2 4/7 
3/5 2/3 5/7 3/4 4/5 5/6 6/7 1/1

Time Complexity of this solution is O(n)

References:
https://en.wikipedia.org/wiki/Farey_sequence

This article is contributed by Utkarsh Trivedi. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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