A259594 - OEIS

A259594

Denominators of the other-side convergents to sqrt(6).

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

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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

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*)