%I #30 Sep 08 2022 08:45:38

%S 0,1,6,13,84,181,1170,2521,16296,35113,226974,489061,3161340,6811741,

%T 44031786,94875313,613283664,1321442641,8541939510,18405321661,

%U 118973869476,256353060613,1657092233154,3570537526921,23080317394680,49731172316281,321467351292366

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

%H Vincenzo Librandi, <a href="/A144535/b144535.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,14,0,-1).

%F From _Colin Barker_, Apr 14 2012: (Start)

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

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

%F 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

%F 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

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

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

%t 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 *)

%t Numerator[Convergents[Sqrt[3]/2,30]] (* or *) LinearRecurrence[{0,14,0,-1},{0,1,6,13},30] (* _Harvey P. Dale_, Feb 10 2014 *)

%o (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

%o (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

%Y Cf. A126664, A144536, A002531/A002530.

%Y Bisections give A001570, A011945.

%K nonn,frac,easy

%O 0,3

%A _N. J. A. Sloane_, Dec 29 2008