A259593 Numerators of the other-side convergents to sqrt(3).

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

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

a(n) = 3^(n/2 - t + 1)*((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}]
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).

Cf. A001075 (even bisection), A005320 (odd bisection).

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