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