A259592 Denominators of the other-side convergents to sqrt(3).

1, 2, 4, 7, 15, 26, 56, 97, 209, 362, 780, 1351, 2911, 5042, 10864, 18817, 40545, 70226, 151316, 262087, 564719, 978122, 2107560, 3650401, 7865521, 13623482, 29354524, 50843527, 109552575, 189750626, 408855776, 708158977, 1525870529, 2642885282, 5694626340

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

Clark Kimberling, 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) = Q(i).

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

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

a(2n) = A001353(n+1); a(2n+1) = A001075(n+1). - Antonio Alberto Olivares, Jul 23 2021

a(n) = 3^(n/2 + 1/2 - t)*((2 + sqrt(3))^t - (-1)^n*(2 - sqrt(3))^t)/2, where t = floor(n/2) + 1. - Ridouane Oudra, Aug 03 2021

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[3]; 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]

Prog

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

Crossrefs

Cf. A002530, A002531, A259593 (numerators).

Cf. A001353 (even bisection), A001075 (odd bisection).

Sequence in context: A358832 A232464 A264292 * A291220 A299099 A248574

Adjacent sequences: A259589 A259590 A259591 * A259593 A259594 A259595

Keyword

nonn,easy,frac

Author

Clark Kimberling, Jul 20 2015

Status

approved