# 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 C++ implementation based on above idea.

 `// C++ program to print Farey Sequence of given order ` `#include ` `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) ` `    ``{ ` `    ``} ` `}; ` ` `  `// Cmparison 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 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; ` `} `

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 ` `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; ` `} `

## 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

 ` `

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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Improved By : jit_t, Mithun Kumar

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