A259588 Denominators of the other-side convergents to e.

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

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 A346268 A097592

Adjacent sequences: A259585 A259586 A259587 * A259589 A259590 A259591

Keyword

nonn,easy,frac

Author

Clark Kimberling, Jul 17 2015

Status

approved