A259591 Numerators of the other-side convergents to Pi.

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

0,1

Comments

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.

Links

Table of n, a(n) for n=0..24.

Formula

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

Example

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

Mathematica

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*)

Crossrefs

Cf. A259590, A002485, A002486.

Sequence in context: A350585 A238510 A222982 * A093959 A308476 A222902

Adjacent sequences: A259588 A259589 A259590 * A259592 A259593 A259594

Keyword

nonn,easy,frac

Author

Clark Kimberling, Jul 17 2015

Status

approved