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A259590 Denominators of the other-side convergents to Pi. 2
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 (list; graph; refs; listen; history; text; internal format)
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], iļ‚®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

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Last modified November 20 10:09 EST 2019. Contains 329334 sequences. (Running on oeis4.)