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A259595 Numerators of the other-side convergents to sqrt(6). 2
3, 7, 27, 71, 267, 703, 2643, 6959, 26163, 68887, 258987, 681911, 2563707, 6750223, 25378083, 66820319, 251217123, 661452967, 2486793147, 6547709351, 24616714347, 64815640543, 243680350323, 641608696079, 2412186788883, 6351271320247, 23878187538507 (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 (0,10,0,-1).

FORMULA

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

a(n) = 10*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 21 2015

G.f.: (x^3-3*x^2+7*x+3) / (x^4-10*x^2+1). - Colin Barker, Jul 21 2015

EXAMPLE

For r = sqrt(6), the first 7 other-side convergents are 3, 7/3, 27/11, 71/29, 267/109, 703/287, 2643/1079. 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    2/1     < sqrt(6) <    3/1               -1

1    5/2     > sqrt(6) >    7/3                1

2    22/9    < sqrt(6) <   27/11              -1

3    49/20   > sqrt(6) >   71/29               1

4    218/89  < sqrt(6) <  267/109             -1

5    485/198 > sqrt(6) >  703/287              1

MATHEMATICA

r = Sqrt[6]; 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]  (* A259594 *)

v = Numerator[t]    (* A259595 *)

PROG

(PARI) Vec((x^3-3*x^2+7*x+3)/(x^4-10*x^2+1) + O(x^50)) \\ Colin Barker, Jul 21 2015

CROSSREFS

Cf. A041006, A041007, A259594.

Sequence in context: A293564 A056257 A066021 * A148742 A148743 A148744

Adjacent sequences:  A259592 A259593 A259594 * A259596 A259597 A259598

KEYWORD

nonn,easy,frac

AUTHOR

Clark Kimberling, Jul 20 2015

STATUS

approved

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Last modified August 21 18:26 EDT 2019. Contains 326168 sequences. (Running on oeis4.)