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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 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

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