A259588 -id:A259588 - cp's OEIS frontend

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

Clark Kimberling, Jul 17 2015

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.

			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

Cf. A259588, A007676, A007677.

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 *)

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

A259590 Denominators of the other-side convergents to Pi.

Original entry on oeis.org

1, 8, 113, 219, 33215, 66317, 99532, 165849, 364913, 630294, 1725033, 3085153, 27235615, 78256779, 131002976, 209259755, 471265707, 1151791169, 2774848045, 6701487259, 11439654911, 574364584667, 1709690779483, 2851718461558, 4561409241041, 47337186164411

Offset: 0

Clark Kimberling, Jul 17 2015

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.

			For r = Pi, the first 7 other-side convergents are 4, 25/8, 355/113, 688/219, 104348/33215, 208341/66317, 312689/99532.
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     3/1    < Pi <    4/1               -1
1    22/7    > Pi >    25/8               1
2   333/106  < Pi <    355/113           -1

Cf. A259591, A002485, A002486, A259588.

r=Pi;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}]
Denominator[t] (* A259590 *)
Numerator[t] (* A259591 *)

cp's OEIS Frontend

A259589 Numerators of the other-side convergents to e.

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