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

Author

Clark Kimberling, Jul 17 2015

Keywords

• nonn
• easy
• frac

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.

Examples

			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
		

Crossrefs

Cf. A259588, A007676, A007677.

Programs

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 *)

Formula

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