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A259589 Numerators of the other-side convergents to e.

%I A259589 #6 Jul 19 2015 11:04:27
%S A259589 3,5,11,19,30,106,193,299,1457,2721,4178,25946,49171,75117,566827,
%T A259589 1084483,1651310,14665106,28245729,42910835,438351041,848456353,
%U A259589 1286807394,14862109042,28875761731,43737870773,563501581931,1098127402131,1661628984062
%N A259589 Numerators of the other-side convergents to e.
%C A259589 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:
%C A259589 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
%C A259589 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.
%C A259589 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.
%F A259589 p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(i).
%e A259589 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.
%e A259589 A comparison of convergents with other-side convergents:
%e A259589 i  p(i)/q(i)       P(i)/Q(i)  p(i)*Q(i) - P(i)*q(i)
%e A259589 0     2/1    < e <    3/1               -1
%e A259589 1     3/1    > e >    5/2                1
%e A259589 2     8/3    < e <   11/4               -1
%e A259589 3    11/4    > e >   19/7                1
%e A259589 4    19/7    < e <   30/11              -1
%e A259589 5    87/32   > e >  106/39               1
%t A259589 r = E; a[i_] := Take[ContinuedFraction[r, 35], i];
%t A259589 b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
%t A259589 t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
%t A259589 u = Denominator[t]  (* A259588 *)
%t A259589 v = Numerator[t]    (* A259589 *)
%Y A259589 Cf. A259588, A007676, A007677.
%K A259589 nonn,easy,frac
%O A259589 0,1
%A A259589 _Clark Kimberling_, Jul 17 2015