# 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 using namespace std; // class for x/y (a term in farey sequenceclass 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 sortingbool cmp(Term a, Term b){    // Comparing two ratio    return a.x * b.y < b.x * a.y;} // GCD of a and bint gcd(int a, int b){    if (b == 0)        return a;    return gcd(b, a % b);} // Function to print Farey sequence of order nvoid farey(int n){    // Create a vector to store terms of output    vector 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 << " ";     // explicitly printing last term    cout << "1/1";} // Driver programint main(){    int n = 7;    cout << "Farey Sequence of order " << n << " is\n";    farey(n);    return 0;}

## Java

 // Java program to print Farey Sequence of given order import java.util.*; // class for x/y (a term in farey sequenceclass Term {    public int x, y;     // Constructor to initialize x and y in x/y    public Term(int x1, int y1)    {        this.x = x1;        this.y = y1;    }}; public class GFG {     // GCD of a and b    static int gcd(int a, int b)    {        if (b == 0)            return a;        return gcd(b, a % b);    }     // Function to print Farey sequence of order n    static void farey(int n)    {        // Create a vector to store terms of output        ArrayList v = new ArrayList();         // 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.add(new Term(i, j));        }         // Sorting the term of sequence        Collections.sort(v, new Comparator() {            // Comparison function for sorting            public int compare(Term a, Term b)            {                // Comparing two ratio                return a.x * b.y - b.x * a.y;            }        });         // Explicitly printing first term        System.out.print("0/1 ");         // Printing other terms        for (int i = 0; i < v.size(); ++i)            System.out.print(v.get(i).x + "/" + v.get(i).y                             + " ");         // explicitly printing last term        System.out.print("1/1");    }     // 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 phasing17

## Python3

 # Python3 program to print# Farey Sequence of given order # class for x/y (a term in farey sequenceclass 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 bdef gcd(a, b):    if b == 0:        return a    return gcd(b, a % b) # Function to print # Farey sequence of order ndef 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 = " ")     # explicitly printing last term    print("1/1") # Driver Codeif __name__ == "__main__":    n = 7    print("Farey sequence of order %d is" % n)    farey(n) # This code is contributed by# sanjeev2552

## C#

 // C# program to print Farey Sequence of given orderusing System;using System.Collections.Generic; class TermComparer : Comparer{     // Comparison function for sorting    public override int Compare(Term a, Term b)    {        // Comparing two ratio        return a.x * b.y - b.x * a.y;    }} // class for x/y (a term in farey sequenceclass Term {    public int x, y;     // Constructor to initialize x and y in x/y    public Term(int x1, int y1)    {        this.x = x1;        this.y = y1;    }}; public class GFG {   // GCD of a and b  static int gcd(int a, int b)  {    if (b == 0)      return a;    return gcd(b, a % b);  }   // Function to print Farey sequence of order n  static void farey(int n)  {    // Create a vector to store terms of output    List v = new List();     // 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.Add(new Term(i, j));    }     // Sorting the term of sequence    v.Sort(new TermComparer());     // Explicitly printing first term    Console.Write("0/1 ");     // Printing other terms    for (int i = 0; i < v.Count; ++i)      Console.Write(v[i].x + "/" + v[i].y + " ");     // explicitly printing last term    Console.Write("1/1");  }   // 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 phasing17

## Javascript



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 xn+2/yn+2 from xn+1/yn+1 and xn/yn

x[n+2] = floor((y[n]+n) / y[n+1])x[n+1]â€“ x[n]
y[n+2] = floor((y[n]+n) / y[n+1])y[n+1]â€“ y[n]

We can use above properties to optimize.

## C++

 // Efficient C++ program to print Farey Sequence of order n#include using namespace std; // Optimized function to print Farey sequence of order nvoid 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 programint 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 nclass GFG{ // Optimized function to print // Farey sequence of order nstatic 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 programpublic 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 nimport math # Optimized function to print Farey # sequence of order ndef 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 Coden = 7;print("Farey Sequence of order", n, "is");farey(n); # This code is contributed by mits

## PHP



## C#

 // Efficient C# program to print // Farey Sequence of order nusing System; public class GFG{  // Optimized function to print // Farey sequence of order nstatic 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 programpublic 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

## Javascript



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

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