A259590 -id:A259590 - cp's OEIS frontend

4, 25, 355, 688, 104348, 208341, 312689, 521030, 1146408, 1980127, 5419351, 9692294, 85563208, 245850922, 411557987, 657408909, 1480524883, 3618458675, 8717442233, 21053343141, 35938735828, 1804419559672, 5371151992734, 8958937768937, 14330089761671

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. A259590, A002485, A002486.

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}]
u = Denominator[t]  (*A259590*)
v = Numerator[t]    (*A259591*)

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

cp's OEIS Frontend

A259591 Numerators of the other-side convergents to Pi.

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