cp's OEIS Frontend

A259588 Denominators of the other-side convergents to e.

%I A259588 #6 Jul 19 2015 11:04:20
%S A259588 1,2,4,7,11,39,71,110,536,1001,1537,9545,18089,27634,208524,398959,
%T A259588 607483,5394991,10391023,15786014,161260336,312129649,473389985,
%U A259588 5467464369,10622799089,16090263458,207300647060,403978495031,611279142091,8690849042711
%N A259588 Denominators of the other-side convergents to e.
%C A259588 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 A259588 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 A259588 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 A259588 Closeness of P(i)/Q(i) to r is indicated by
%C A259588 |r - P(i)/Q(i)| < |p(i)/q(i) - P(i)/Q(i)| = 1/(q(i)Q(i)), for i >= 0.
%F A259588 p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = Q(i).
%e A259588 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 A259588 A comparison of convergents with other-side convergents:
%e A259588 i    p(i)/q(i)      P(i)/Q(i)  p(i)Q(i)-P(i)q(i)
%e A259588 0       2/1   < e <   3/1              -1
%e A259588 1       3/1   > e >   5/2               1
%e A259588 2       8/3   < e <   11/4             -1
%e A259588 3      11/4   > e >   19/7              1
%e A259588 4      19/7   < e <   30/11            -1
%e A259588 5      87/32  > e >  106/39             1
%t A259588 r = E; a[i_] := Take[ContinuedFraction[r, 35], i];
%t A259588 b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
%t A259588 t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
%t A259588 u = Denominator[t]  (* A259588 *)
%t A259588 v = Numerator[t]    (* A259589 *)
%Y A259588 Cf. A259589, A007676, A007677.
%K A259588 nonn,easy,frac
%O A259588 0,2
%A A259588 _Clark Kimberling_, Jul 17 2015