Farey sequence
- Farey sequence
(sometimes incorrectly called a Farey series.)
- definition
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.
Each Farey sequence starts with value 0, by the fraction and ends with the value 1, denoted by the fraction .
The Farey sequences of orders 1 to 5 are:
- F1 =
- F2 =
- F3 =
- F4 =
- F5 =
- task requirements
- compute and show the Farey sequence for order 1 through 11 (inclusive).
- compute and display the number of fractions in the Farey sequence for order 100 through 1,000 (inclusive) by hundreds.
- see also
- sequence A006842 numerators of Farey series of order 1, 2, ··· on OEIS, The On-line Encyclopedia of Integer Sequences (R).
- sequence A006843 denominators of Farey series of order 1, 2, ··· on OEIS, The On-line Encyclopedia of Integer Sequences (R).
- sequence A005728 number of fractions in Farey series of order n. on OEIS, The On-line Encyclopedia of Integer Sequences (R).
- entry Farey sequence on Wolfram Mathworld (TM).
- Wiki Farey sequence in Wikipedia.
REXX
<lang rexx>/*REXX program computes & shows a Farey sequence (or the # of fractions)*/ parse arg L H I . /*get optional values from C.L. */ if L== then L=5 /*L not specified? Use default.*/ oldL=L /*original L (negativity=noshow)*/ L=abs(L) /*but ··· use |L| for all else.*/ if H== then H=L /*H not specified? Use default.*/ if I== then I=1 /*I " " " " */
do n=L to H by I /*process range (could be only 1)*/ @=fareyF(n); #=' 'words(@)" " /*go ye forth & compute Farey #s.*/ say center('Farey sequence for: 1/'n " has" # 'fractions.',150,'═') if oldL>0 then say @; say /*display Farey fractions (if +).*/ end /*n*/ /* [↑] build/show Farey fractions*/
exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────FAREYF subroutine───────────────────*/ fareyf: procedure; parse arg x; n.1=0; d.1=1; n.2=1; d.2=x /*kit parts.*/ $=n.1'/'d.1 n.2"/"d.2 /*a starter kit for the Farey seq*/
/* [↓] now, build on the starter*/ do j=1; y=j+1; z=j+2 /*construct from thirds on "up". */ n.z=((d.j+x)%d.y)*n.y-n.j /* " fraction numerator. */ d.z=((d.j+x)%d.y)*d.y-d.j /* " " denominator.*/ if n.z>x then leave /*Should construction be stopped?*/ $=$ n.z'/'d.z /*Heck no, add this to party mix.*/ end /*j*/ /* [↑] construct the Farey seq. */
return $ /*return with the fractions. */</lang> output when using the following for input: 1 11
═════════════════════════════════════════════════════Farey sequence for: 1/1 has 2 fractions.═════════════════════════════════════════════════════ 0/1 1/1 ═════════════════════════════════════════════════════Farey sequence for: 1/2 has 3 fractions.═════════════════════════════════════════════════════ 0/1 1/2 1/1 ═════════════════════════════════════════════════════Farey sequence for: 1/3 has 5 fractions.═════════════════════════════════════════════════════ 0/1 1/3 1/2 2/3 1/1 ═════════════════════════════════════════════════════Farey sequence for: 1/4 has 7 fractions.═════════════════════════════════════════════════════ 0/1 1/4 1/3 1/2 2/3 3/4 1/1 ════════════════════════════════════════════════════Farey sequence for: 1/5 has 11 fractions.═════════════════════════════════════════════════════ 0/1 1/5 1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 1/1 ════════════════════════════════════════════════════Farey sequence for: 1/6 has 13 fractions.═════════════════════════════════════════════════════ 0/1 1/6 1/5 1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 5/6 1/1 ════════════════════════════════════════════════════Farey sequence for: 1/7 has 19 fractions.═════════════════════════════════════════════════════ 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 for: 1/8 has 23 fractions.═════════════════════════════════════════════════════ 0/1 1/8 1/7 1/6 1/5 1/4 2/7 1/3 3/8 2/5 3/7 1/2 4/7 3/5 5/8 2/3 5/7 3/4 4/5 5/6 6/7 7/8 1/1 ════════════════════════════════════════════════════Farey sequence for: 1/9 has 29 fractions.═════════════════════════════════════════════════════ 0/1 1/9 1/8 1/7 1/6 1/5 2/9 1/4 2/7 1/3 3/8 2/5 3/7 4/9 1/2 5/9 4/7 3/5 5/8 2/3 5/7 3/4 7/9 4/5 5/6 6/7 7/8 8/9 1/1 ════════════════════════════════════════════════════Farey sequence for: 1/10 has 33 fractions.════════════════════════════════════════════════════ 0/1 1/10 1/9 1/8 1/7 1/6 1/5 2/9 1/4 2/7 3/10 1/3 3/8 2/5 3/7 4/9 1/2 5/9 4/7 3/5 5/8 2/3 7/10 5/7 3/4 7/9 4/5 5/6 6/7 7/8 8/9 9/10 1/1 ════════════════════════════════════════════════════Farey sequence for: 1/11 has 43 fractions.════════════════════════════════════════════════════ 0/1 1/11 1/10 1/9 1/8 1/7 1/6 2/11 1/5 2/9 1/4 3/11 2/7 3/10 1/3 4/11 3/8 2/5 3/7 4/9 5/11 1/2 6/11 5/9 4/7 3/5 5/8 7/11 2/3 7/10 5/7 8/11 3/4 7/9 4/5 9/11 5/6 6/7 7/8 8/9 9/10 10/11 1/1
output when using the following for input: -100 1000 100
═════════════════════════════════════════════════Farey sequence for: 1/100 has 3045 fractions. ═══════════════════════════════════════════════════ ═════════════════════════════════════════════════Farey sequence for: 1/200 has 12233 fractions. ═══════════════════════════════════════════════════ ═════════════════════════════════════════════════Farey sequence for: 1/300 has 27399 fractions. ═══════════════════════════════════════════════════ ═════════════════════════════════════════════════Farey sequence for: 1/400 has 48679 fractions. ═══════════════════════════════════════════════════ ═════════════════════════════════════════════════Farey sequence for: 1/500 has 76117 fractions. ═══════════════════════════════════════════════════ ════════════════════════════════════════════════Farey sequence for: 1/600 has 109501 fractions. ═══════════════════════════════════════════════════ ════════════════════════════════════════════════Farey sequence for: 1/700 has 149019 fractions. ═══════════════════════════════════════════════════ ════════════════════════════════════════════════Farey sequence for: 1/800 has 194751 fractions. ═══════════════════════════════════════════════════ ════════════════════════════════════════════════Farey sequence for: 1/900 has 246327 fractions. ═══════════════════════════════════════════════════ ════════════════════════════════════════════════Farey sequence for: 1/1000 has 304193 fractions. ═══════════════════════════════════════════════════