40 lines
2.6 KiB
Text
40 lines
2.6 KiB
Text
The [[wp:Farey sequence|Farey sequence]] ''' ''F''<sub>n</sub>''' 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.
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The ''Farey sequence'' is sometimes incorrectly called a ''Farey series''.
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Each Farey sequence:
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:::* starts with the value '''0''' (zero), denoted by the fraction <math> \frac{0}{1} </math>
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:::* ends with the value '''1''' (unity), denoted by the fraction <math> \frac{1}{1}</math>.
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The Farey sequences of orders '''1''' to '''5''' are:
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:::: <math>{\bf\it{F}}_1 = \frac{0}{1}, \frac{1}{1}</math>
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:<br>
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:::: <math>{\bf\it{F}}_2 = \frac{0}{1}, \frac{1}{2}, \frac{1}{1}</math>
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:<br>
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:::: <math>{\bf\it{F}}_3 = \frac{0}{1}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{1}{1}</math>
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:<br>
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:::: <math>{\bf\it{F}}_4 = \frac{0}{1}, \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{1}{1}</math>
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:<br>
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:::: <math>{\bf\it{F}}_5 = \frac{0}{1}, \frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{1}{1}</math>
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;Task
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* Compute and show the Farey sequence for orders '''1''' through '''11''' (inclusive).
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* Compute and display the ''number'' of fractions in the Farey sequence for order '''100''' through '''1,000''' (inclusive) by hundreds.
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* Show the fractions as <big> '''n/d''' </big> (using the solidus [or slash] to separate the numerator from the denominator).
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The length (the number of fractions) of a Farey sequence asymptotically approaches:
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::::::::::: <big><big> 3 × n<sup>2</sup> <big> ÷ </big> <big><math>\pi</math></big><sup>2</sup> </big></big>
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;See also:
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* OEIS sequence [[oeis:A006842|A006842 numerators of Farey series of order 1, 2, ···]]
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* OEIS sequence [[oeis:A006843|A006843 denominators of Farey series of order 1, 2, ···]]
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* OEIS sequence [[oeis:A005728|A005728 number of fractions in Farey series of order n]]
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* MathWorld entry [http://mathworld.wolfram.com/FareySequence.html Farey sequence]
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* Wikipedia entry [[wp:Farey_sequence|Farey sequence]]
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<br><br>
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