How to resolve the algorithm Farey sequence step by step in the Maxima programming language

Published on 12 May 2024 09:40 PM
#Maxima

How to resolve the algorithm Farey sequence step by step in the Maxima programming language

Table of Contents

Problem Statement

The   Farey sequence   Fn   of order   n   is the sequence of completely reduced fractions between   0   and   1   which, when in lowest terms, have denominators less than or equal to   n,   arranged in order of increasing size. The   Farey sequence   is sometimes incorrectly called a   Farey series.

Each Farey sequence:

The Farey sequences of orders   1   to   5   are:

The length   (the number of fractions)   of a Farey sequence asymptotically approaches:

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Farey sequence step by step in the Maxima programming language

Source code in the maxima programming language

farey(n):=if n=1 then ["0/1","1/1"] else block(
    create_list([i,j],i,0,n-1,j,1,n),
    map(lambda([x],if x[1]=0 and x[2]#1 then false else if x[1]=x[2] and x[1]#1 then false else if x[1]<=x[2] then x),%%),
    delete(false,%%),
    unique(map(lambda([x],x[1]/x[2]),%%)),
    append(rest(append(["0/1"],rest(%%)),-1),["1/1"])
)$

/* Test cases */
/* Sequences from order 1 through 11 */
farey(1);
farey(2);
farey(3);
farey(4);
farey(5);
farey(6);
farey(7);
farey(8);
farey(9);
farey(10);
farey(11);

/* Length of sequences from order 100 through, 1000 by hundreds */
length(farey(100));
length(farey(200));
length(farey(300));
length(farey(400));
length(farey(500));
length(farey(600));
length(farey(700));
length(farey(800));
length(farey(900));
length(farey(1000));

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