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A259594 Denominators of the other-side convergents to sqrt(6). 2
1, 3, 11, 29, 109, 287, 1079, 2841, 10681, 28123, 105731, 278389, 1046629, 2755767, 10360559, 27279281, 102558961, 270037043, 1015229051, 2673091149, 10049731549, 26460874447, 99482086439, 261935653321, 984771132841, 2592895658763, 9748229241971 (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

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) = Q(i).

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

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

Closed form: a(n) = (((-1)^n-5/2)*sqrt(6)/6+1-1/2*( -1)^n)*(5-2*sqrt(6))^(-1/4)*(5-2*sqrt(6))^(1/2*n)*(5-2*sqrt(6))^(1/4*(-1)^n)+((5 /2-(-1)^n)*sqrt(6)/6+1-1/2 *(-1)^n)*(5+2*sqrt(6))^(1/2*n)*(5+2*sqrt(6))^(1/4*(-1)^n)*(5+2*sqrt(6))^(-1/4). - Paolo P. Lava, Aug 05 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+1)*(x^2-2*x-1)/(x^4-10*x^2+1) + O(x^50)) \\ Colin Barker, Jul 21 2015

CROSSREFS

Cf. A041006, A041007, A259595.

Sequence in context: A159229 A239713 A122023 * A293010 A236467 A328550

Adjacent sequences:  A259591 A259592 A259593 * A259595 A259596 A259597

KEYWORD

nonn,easy,frac

AUTHOR

Clark Kimberling, Jul 20 2015

STATUS

approved

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Last modified December 4 19:40 EST 2021. Contains 349526 sequences. (Running on oeis4.)