The On-Line Encyclopedia of Integer Sequences (OEIS)

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A097545

Numerators of "Farey fraction" approximations to Pi.
(history; published version)

#21 by Bruno Berselli at Mon Sep 06 11:54:16 EDT 2021

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#20 by Michel Marcus at Mon Sep 06 11:40:30 EDT 2021

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#19 by Jon E. Schoenfield at Mon Sep 06 11:09:15 EDT 2021

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#18 by Jon E. Schoenfield at Mon Sep 06 11:09:08 EDT 2021

COMMENTS

Write a 0 if f_n <= x and a 1 if f_n > x. This gives (for x = Pi) the sequence 1, 0, 0, 0, 1, 1, 1, 1, 0 (7 times), 1 (15 times, ), 0, 1, ... Ignore the initial string 1, 0, 0, 0, which is always the same. Look at the runs run lengths of the remaining sequence, which are in this case L_1 = 4, L_2 = 7, L_3 = 15, L_4 = 1, L_5 = 292, etc. (A001203). Christoffel showed that x has the continued fraction representation (L_1 - 1) + 1/(L_2 + 1/(L_3 + 1/(L_4 + ...))).

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#17 by Alois P. Heinz at Fri Jul 07 10:23:59 EDT 2017

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#16 by Michel Marcus at Fri Jul 07 10:22:58 EDT 2017

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#15 by Michel Marcus at Fri Jul 07 10:22:55 EDT 2017

REFERENCES

C. Brezinski, History of Continued Fractions and Pade' Padé Approximants, Springer-Verlag, 1991; pp. 151-152.

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#14 by N. J. A. Sloane at Sat May 07 12:29:18 EDT 2016

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#13 by N. J. A. Sloane at Sat May 07 12:29:10 EDT 2016

LINKS

Dave Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/99/farey">Farey fractions on sci.math</a> [Broken link]

Dave Rusin, <a href="/A002965/a002965.txt">Farey fractions on sci.math</a> [Cached copy]

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#12 by Reinhard Zumkeller at Mon Jan 27 15:19:29 EST 2014

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approved