A144535 Numerators of continued fraction convergents to sqrt(3)/2.

0, 1, 6, 13, 84, 181, 1170, 2521, 16296, 35113, 226974, 489061, 3161340, 6811741, 44031786, 94875313, 613283664, 1321442641, 8541939510, 18405321661, 118973869476, 256353060613, 1657092233154, 3570537526921, 23080317394680, 49731172316281, 321467351292366

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (0,14,0,-1).

Formula

From Colin Barker, Apr 14 2012: (Start)

a(n) = 14*a(n-2) - a(n-4).

G.f.: x*(1 + 6*x - x^2)/((1 - 4*x + x^2)*(1 + 4*x + x^2)). (End)

a(n) = ((-(-2-sqrt(3))^n*(-3+sqrt(3)) + (2-sqrt(3))^n*(-3+sqrt(3)) - (3+sqrt(3))*((-2+sqrt(3))^n - (2+sqrt(3))^n)))/(8*sqrt(3)). - Colin Barker, Mar 27 2016

a(2*n) = 6*a(2*n-1) + a(2*n-2). a(2*n+1) = A003154(A101265(n+1)). - John Elias, Dec 10 2021

Example

0, 1, 6/7, 13/15, 84/97, 181/209, 1170/1351, 2521/2911, 16296/18817, 35113/40545, ...

Maple

with(numtheory); Digits:=200: cf:=convert(evalf(sqrt(3)/2, confrac); [seq(nthconver(cf, i), i=0..100)];

Mathematica

CoefficientList[Series[x (1 + 6 x - x^2)/((1 - 4 x + x^2) (1 + 4 x + x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 10 2013 *)
Numerator[Convergents[Sqrt[3]/2, 30]] (* or *) LinearRecurrence[{0, 14, 0, -1}, {0, 1, 6, 13}, 30] (* Harvey P. Dale, Feb 10 2014 *)

Prog

(Magma) I:=[0, 1, 6, 13]; [n le 4 select I[n] else 14*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013
(PARI) Vec(x*(1+6*x-x^2)/((1-4*x+x^2)*(1+4*x+x^2)) + O(x^30)) \\ Colin Barker, Mar 27 2016

Crossrefs

Cf. A126664, A144536, A002531/A002530.

Bisections give A001570, A011945.

Sequence in context: A119110 A041305 A215755 * A042641 A292121 A364199

Adjacent sequences: A144532 A144533 A144534 * A144536 A144537 A144538

Keyword

nonn,frac,easy

Author

N. J. A. Sloane, Dec 29 2008

Status

approved