A259590 Denominators of the other-side convergents to Pi.

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

0,2

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..25.

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}]
Denominator[t] (* A259590 *)
Numerator[t] (* A259591 *)

Crossrefs

Cf. A259591, A002485, A002486, A259588.

Sequence in context: A099703 A296467 A164774 * A155460 A064090 A072402

Adjacent sequences: A259587 A259588 A259589 * A259591 A259592 A259593

Keyword

nonn,easy,frac

Author

Clark Kimberling, Jul 17 2015

Status

approved