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A259588 Denominators of the other-side convergents to e. 3
1, 2, 4, 7, 11, 39, 71, 110, 536, 1001, 1537, 9545, 18089, 27634, 208524, 398959, 607483, 5394991, 10391023, 15786014, 161260336, 312129649, 473389985, 5467464369, 10622799089, 16090263458, 207300647060, 403978495031, 611279142091, 8690849042711 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Suppose that a positive irrational number r has continued fraction [a(0), a(1), ... ]. Define sequences p(i), q(i), P(i), Q(i) from the numerators and denominators of finite continued fractions as follows:

p(i)/q(i) = [a(0), a(1), ... a(i)] and P(i)/Q(i) = [a(0), a(1), ..., a(i) + 1]. The fractions p(i)/q(i) are the convergents to r, and the fractions P(i)/Q(i) are introduced here as the "other-side convergents" to

r, because p(2k)/q(2k) < r < P(2k)/Q(2k) and P(2k+1)/Q(2k+1) < r < p(2k+1)/q(2k+1), for k >= 0.

Closeness of P(i)/Q(i) to r is indicated by

|r - P(i)/Q(i)| < |p(i)/q(i) - P(i)/Q(i)| = 1/(q(i)Q(i)), for i >= 0.

LINKS

Table of n, a(n) for n=0..29.

FORMULA

p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = Q(i).

EXAMPLE

For r = e, the first 13 other-side convergents are 3/1, 5/2, 11/4, 19/7, 30/11, 106/39, 193/71, 299/110, 1457/536, 2721/1001, 4178/1537, 25946/9545, 49171/18089.

A comparison of convergents with other-side convergents:

i    p(i)/q(i)      P(i)/Q(i)  p(i)Q(i)-P(i)q(i)

0       2/1   < e <   3/1              -1

1       3/1   > e >   5/2               1

2       8/3   < e <   11/4             -1

3      11/4   > e >   19/7              1

4      19/7   < e <   30/11            -1

5      87/32  > e >  106/39             1

MATHEMATICA

r = E; a[i_] := Take[ContinuedFraction[r, 35], i];

b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];

t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]

u = Denominator[t]  (* A259588 *)

v = Numerator[t]    (* A259589 *)

CROSSREFS

Cf. A259589, A007676, A007677.

Sequence in context: A261145 A277339 A153555 * A058103 A097592 A267699

Adjacent sequences:  A259585 A259586 A259587 * A259589 A259590 A259591

KEYWORD

nonn,easy,frac

AUTHOR

Clark Kimberling, Jul 17 2015

STATUS

approved

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Last modified June 28 03:08 EDT 2017. Contains 288813 sequences.