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 << " " ; // explicitly 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; } |
Java
// Java program to print Farey Sequence of given order import java.util.*; // class for x/y (a term in farey sequence class 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<Term> v = new ArrayList<Term>(); // 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<Term>() { // 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 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 = " " ) # explicitly 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 |
C#
// C# program to print Farey Sequence of given order using System; using System.Collections.Generic; class TermComparer : Comparer<Term> { // 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 sequence class 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<Term> v = new List<Term>(); // 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
<script> // Javascript 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 constructor(x, y) { this .x = x; this .y = y; } } // GCD of a and b function gcd(a, b) { if (b == 0) return a return gcd(b, a % b) } // Function to print // Farey sequence of order n function farey(n) { // Create a vector to // store terms of output let v = [] // One by one find and store // all terms except 0/1 and n/n // which are known for (let i=1;i<n + 1;i++) { for (j=i+1;j<n + 1;j++) { // Checking whether i and j // are in lowest term if (gcd(i, j) == 1) v.push( new Term(i, j)) } } // Sorting the term of sequence for (let i=0;i<(v).length;i++) { for (let j=i + 1; j<(v).length; j++) { if (v[i].x * v[j].y > v[j].x * v[i].y) { let temp = v[j]; v[j] = v[i]; v[i] = temp; } } } // Explicitly printing first term document.write( "0/1 " ) // Printing other terms for (let i=0;i<(v).length;i++) document.write(v[i].x+ "/" +v[i].y+ " " ) // explicitly printing last term document.write( "1/1" ) } // Driver Code let n = 7; document.write( "Farey sequence of order " +n+ " is<br>" ); farey(n) // This code is contributed by patel2127 </script> |
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]– 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 |
Javascript
<script> // Efficient javascript 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 var x1 = 0, y1 = 1, x2 = 1, y2 = n; document.write(x1+ "/" +y1+ " " +x2+ "/" +y2+ " " ); var 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 document.write(x+ "/" + y+ " " ); // Update x1, y1, x2 and y2 for next iteration x1 = x2; x2 = x; y1 = y2; y2 = y; } } // Driver program var n = 7; document.write( "Farey Sequence of order " + n + " is<br/>" ); farey(n); // This code is contributed by gauravrajput1 </script> |
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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