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A097546
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Denominators of "Farey fraction" approximations to Pi.
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6
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0, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78, 85, 92, 99, 106, 113, 219, 332, 445, 558, 671, 784, 897, 1010, 1123, 1236, 1349, 1462, 1575, 1688, 1801, 1914, 2027, 2140, 2253, 2366, 2479, 2592, 2705, 2818, 2931, 3044, 3157, 3270
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OFFSET
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0,7
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COMMENTS
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Given a real number x >= 1 (here x = Pi), start with 1/0 and 0/1 and construct the sequence of fractions f_n = r_n/s_n such that:
f_{n+1} = (r_k + r_n)/(s_k + s_n) where k is the greatest integer < n such that f_k <= x <= f_n. Sequence gives values s_n.
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 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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REFERENCES
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C. Brezinski, History of Continued Fractions and Padé Approximants, Springer-Verlag, 1991; pp. 151-152.
E. B. Christoffel, Observatio arithmetica, Ann. Math. Pura Appl., (II) 6 (1875), 148-153.
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LINKS
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EXAMPLE
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The fractions are 1/0, 0/1, 1/1, 2/1, 3/1, 4/1, 7/2, 10/3, 13/4, 16/5, 19/6, 22/7, 25/8, 47/15, ...
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MATHEMATICA
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f[x_, n_] := (m = Floor[x]; f0 = {m, m+1/2, m+1};
r = ({a___, b_, c_, d___} /; b < x < c) :> {b, (Numerator[b] + Numerator[c]) / (Denominator[b] + Denominator[c]), c}; Join[{m, m+1}, NestList[# /. r &, f0, n-3][[All, 2]]]); Join[{1, 0, 1, 2}, f[Pi, 52]] // Denominator (* Jean-François Alcover, May 18 2011 *)
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CROSSREFS
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KEYWORD
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nonn,frac,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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