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=={{header|J}}==
'''Solution:'''
<lang J>Farey=: x:@/:~@(0 , ~.)@(#~ <:&1)@:,@(%/~@(1 + i.)) NB. calculates Farey sequence
displayFarey=: ('r/' charsub '0r' , ,&'r1')@": NB. displays Farey sequence according to task requirements
order=: ': ' ,~ ": NB. display order of Farey sequence</lang>
'''Required examples:'''
{{improve|J| <br><br> The output for the first and last term (as per the task's requirement)
<br> is to show the first term as <big>'''0/1'''</big>,
<br> and to show the last term as <big>'''1/1'''</big>.
<br> Also, please ''translate'' the '''r''' character to a solidus if possible. <br><br> }}
<lang J> LF joinstring (order , displayFarey@Farey)&.> 1 + i.11 NB. Farey sequences, order 1-11
1: 0/0 1/1
J has an internal data representation for completely reduced rational numbers. This displays as integers where that is possible and otherwise displays as NNNrDDD where the part to the left of the 'r' is the numerator and the part to the right of the 'r' is the denominator.
2: 0/0 1/2 1/1
3: 0/0 1/3 1/2 2/3 1/1
This mechanism is a part of J's "constant language", and is similar to scientific notation (which uses an 'e' instead of an 'r') and with J's complex number notation (which uses a 'j' instead of an 'r'), and which follow similar display rules.
4: 0/0 1/4 1/3 1/2 2/3 3/4 1/1
5: 0/0 1/5 1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 1/1
This mechanism also hints that J's type promotion rules are designed to give internally consistent results a priority. As much as possible you do not get different results from the same operation just because you "used a different data type". J's design adopts the philosophy that "different results from the same operation based on different types" is likely to introduce errors in thinking. (Of course there are machine limits and certain floating point operations tend to introduce internal inconsistencies, but those are mentioned only in passing - they are not directly relevant to this task.)
6: 0/0 1/6 1/5 1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 5/6 1/1
7: 0/0 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
<lang J>Farey=:3 :0
8: 0/0 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
0,/:~~.(#~ <:&1),%/~1x+i.y
9: 0/0 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
)</lang>
10: 0/0 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
11: 0/0 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
Required examples:
LF joinstring (order , ":@#@Farey)&.> 100 * 1 + i.10 NB. Count of Farey sequence items, order 100,200,..1000
100: 3045
<lang J> Farey 1
200: 12233
0 1
300: 27399
Farey 2
400: 48679
0 1r2 1
500: 76117
Farey 3
600: 109501
0 1r3 1r2 2r3 1
700: 149019
Farey 4
800: 194751
0 1r4 1r3 1r2 2r3 3r4 1
900: 246327
Farey 5
1000: 304193</lang>
0 1r5 1r4 1r3 2r5 1r2 3r5 2r3 3r4 4r5 1
Farey 6
0 1r6 1r5 1r4 1r3 2r5 1r2 3r5 2r3 3r4 4r5 5r6 1
Farey 7
0 1r7 1r6 1r5 1r4 2r7 1r3 2r5 3r7 1r2 4r7 3r5 2r3 5r7 3r4 4r5 5r6 6r7 1
Farey 8
0 1r8 1r7 1r6 1r5 1r4 2r7 1r3 3r8 2r5 3r7 1r2 4r7 3r5 5r8 2r3 5r7 3r4 4r5 5r6 6r7 7r8 1
Farey 9
0 1r9 1r8 1r7 1r6 1r5 2r9 1r4 2r7 1r3 3r8 2r5 3r7 4r9 1r2 5r9 4r7 3r5 5r8 2r3 5r7 3r4 7r9 4r5 5r6 6r7 7r8 8r9 1
Farey 10
0 1r10 1r9 1r8 1r7 1r6 1r5 2r9 1r4 2r7 3r10 1r3 3r8 2r5 3r7 4r9 1r2 5r9 4r7 3r5 5r8 2r3 7r10 5r7 3r4 7r9 4r5 5r6 6r7 7r8 8r9 9r10 1
Farey 11
0 1r11 1r10 1r9 1r8 1r7 1r6 2r11 1r5 2r9 1r4 3r11 2r7 3r10 1r3 4r11 3r8 2r5 3r7 4r9 5r11 1r2 6r11 5r9 4r7 3r5 5r8 7r11 2r3 7r10 5r7 8r11 3r4 7r9 4r5 9r11 5r6 6r7 7r8 8r9 9r10 10r11 1
(,. #@Farey"0) 100*1+i.10
100 3045
200 12233
300 27399
400 48679
500 76117
600 109501
700 149019
800 194751
900 246327
1000 304193</lang>
=== Optimized ===
A small change in the 'Farey' function makes the last request, faster.
A second change in the 'Farey' function makes the last request, much faster.
A third change in the 'Farey' function makes the last request, again, a little bit faster.
<strike>Even if it is 20 times faster, the response time is just acceptable.</strike>
Now the response time is quite satisfactory.
The script produces the sequences in rational number notation as well in fractional number notation.
<lang J>Farey=: 3 : '/:~,&0 1~.(#~<&1),(1&+%/2&+)i.y-1'
NB. rational number notation
rplc&(' 0';'= 0r0');,&('r1',LF)@:,~&'F'@:":@:x:&.>(,Farey)&.>1+i.11
NB. fractional number notation
rplc&('r';'/';' 0';'= 0/0');,&('r1',LF)@:,~&'F'@:":@:x:&.>(,Farey)&.>1+i.11
NB. number of fractions
;,&(' items',LF)@:,~&'F'@:":&.>(,.#@:Farey)&.>100*1+i.10</lang>
{{out}}
<pre>
F1= 0r0 1r1
F2= 0r0 1r2 1r1
F3= 0r0 1r3 1r2 2r3 1r1
F4= 0r0 1r4 1r3 1r2 2r3 3r4 1r1
F5= 0r0 1r5 1r4 1r3 2r5 1r2 3r5 2r3 3r4 4r5 1r1
F6= 0r0 1r6 1r5 1r4 1r3 2r5 1r2 3r5 2r3 3r4 4r5 5r6 1r1
F7= 0r0 1r7 1r6 1r5 1r4 2r7 1r3 2r5 3r7 1r2 4r7 3r5 2r3 5r7 3r4 4r5 5r6 6r7 1r1
F8= 0r0 1r8 1r7 1r6 1r5 1r4 2r7 1r3 3r8 2r5 3r7 1r2 4r7 3r5 5r8 2r3 5r7 3r4 4r5 5r6 6r7 7r8 1r1
F9= 0r0 1r9 1r8 1r7 1r6 1r5 2r9 1r4 2r7 1r3 3r8 2r5 3r7 4r9 1r2 5r9 4r7 3r5 5r8 2r3 5r7 3r4 7r9 4r5 5r6 6r7 7r8 8r9 1r1
F10= 0r0 1r10 1r9 1r8 1r7 1r6 1r5 2r9 1r4 2r7 3r10 1r3 3r8 2r5 3r7 4r9 1r2 5r9 4r7 3r5 5r8 2r3 7r10 5r7 3r4 7r9 4r5 5r6 6r7 7r8 8r9 9r10 1r1
F11= 0r0 1r11 1r10 1r9 1r8 1r7 1r6 2r11 1r5 2r9 1r4 3r11 2r7 3r10 1r3 4r11 3r8 2r5 3r7 4r9 5r11 1r2 6r11 5r9 4r7 3r5 5r8 7r11 2r3 7r10 5r7 8r11 3r4 7r9 4r5 9r11 5r6 6r7 7r8 8r9 9r10 10r11 1r1
F1= 0/0 1/1
F2= 0/0 1/2 1/1
F3= 0/0 1/3 1/2 2/3 1/1
F4= 0/0 1/4 1/3 1/2 2/3 3/4 1/1
F5= 0/0 1/5 1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 1/1
F6= 0/0 1/6 1/5 1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 5/6 1/1
F7= 0/0 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
F8= 0/0 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
F9= 0/0 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
F10= 0/0 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
F11= 0/0 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
F100 3045 items
F200 12233 items
F300 27399 items
F400 48679 items
F500 76117 items
F600 109501 items
F700 149019 items
F800 194751 items
F900 246327 items
F1000 304193 items
</pre>
=={{header|Java}}==
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