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 A259593 Numerators of the other-side convergents to sqrt(3). 2
 2, 3, 7, 12, 26, 45, 97, 168, 362, 627, 1351, 2340, 5042, 8733, 18817, 32592, 70226, 121635, 262087, 453948, 978122, 1694157, 3650401, 6322680, 13623482, 23596563, 50843527, 88063572, 189750626, 328657725, 708158977, 1226567328, 2642885282, 4577611587 (list; graph; refs; listen; history; text; internal format)
 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 Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1). FORMULA p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(i). a(n) = 4*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 21 2015 G.f.: -(x^2-3*x-2) / (x^4-4*x^2+1). - Colin Barker, Jul 21 2015 EXAMPLE For r = sqrt(3), 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      1/1  < sqrt(3) <     2/1               -1 1      2/1  > sqrt(3) >     3/2                1 2      5/3  < sqrt(3) <     7/4               -1 3      7/4  > sqrt(3) >    12/7                1 4     19/11 < sqrt(3) <    26/15              -1 5     26/15 > sqrt(3) >    45/26               1 MATHEMATICA r = Sqrt; 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}] v = Numerator[t] PROG (PARI) Vec(-(x^2-3*x-2)/(x^4-4*x^2+1) + O(x^50)) \\ Colin Barker, Jul 21 2015 CROSSREFS Cf. A002530, A002531, A259592 (denominators). Sequence in context: A321838 A298897 A054272 * A129016 A099163 A000676 Adjacent sequences:  A259590 A259591 A259592 * A259594 A259595 A259596 KEYWORD nonn,easy,frac AUTHOR Clark Kimberling, Jul 20 2015 STATUS approved

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Last modified August 20 18:56 EDT 2019. Contains 326154 sequences. (Running on oeis4.)