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A144535 Numerators of continued fraction convergents to sqrt(3)/2. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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)). - Colin Barker, Apr 14 2012

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

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 A236250

Adjacent sequences:  A144532 A144533 A144534 * A144536 A144537 A144538

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane, Dec 29 2008

STATUS

approved

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Last modified July 22 10:41 EDT 2019. Contains 325219 sequences. (Running on oeis4.)