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
Examples
0, 1, 6/7, 13/15, 84/97, 181/209, 1170/1351, 2521/2911, 16296/18817, 35113/40545, ...
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).
Programs
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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
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Maple
with(numtheory); Digits:=200: cf:=convert(evalf(sqrt(3)/2,confrac); [seq(nthconver(cf,i), i=0..100)];
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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 *) -
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
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