A259589 Numerators of the other-side convergents to e.

3, 5, 11, 19, 30, 106, 193, 299, 1457, 2721, 4178, 25946, 49171, 75117, 566827, 1084483, 1651310, 14665106, 28245729, 42910835, 438351041, 848456353, 1286807394, 14862109042, 28875761731, 43737870773, 563501581931, 1098127402131, 1661628984062

Offset

0,1

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..28.

Formula

p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(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. A259588, A007676, A007677.

Sequence in context: A284034 A051642 A007671 * A090471 A124098 A127453

Adjacent sequences: A259586 A259587 A259588 * A259590 A259591 A259592

Keyword

nonn,easy,frac

Author

Clark Kimberling, Jul 17 2015

Status

approved