A041019 Denominators of continued fraction convergents to sqrt(13).

1, 1, 2, 3, 5, 33, 38, 71, 109, 180, 1189, 1369, 2558, 3927, 6485, 42837, 49322, 92159, 141481, 233640, 1543321, 1776961, 3320282, 5097243, 8417525, 55602393, 64019918, 119622311, 183642229, 303264540, 2003229469, 2306494009, 4309723478, 6616217487, 10925940965

Offset

0,3

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,36,0,0,0,0,1).

Formula

From Johannes W. Meijer, Jun 12 2010: (Start)

a(5*n) = A006190(3*n+1),

a(5*n+1) = (A006190(3*n+2) - A006190(3*n+1))/2,

a(5*n+2) = (A006190(3*n+2) + A006190(3*n+1))/2,

a(5*n+3) = A006190(3*n+2) and a(5*n+4) = A006190(3*n+3)/2. (End)

G.f.: ((1 - 2*x + 4*x^2 - 3*x^3 + x^4)*(1 + 3*x + 4*x^2 + 2*x^3 + x^4))/(1 - 36*x^5 - x^10). - Peter J. C. Moses, Jul 29 2013

a(n) = A010122(n)*a(n-1) + a(n-2), a(0)=1, a(-1)=0. - Paul Weisenhorn, Aug 17 2018

Mathematica

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[13], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
CoefficientList[Series[((1 - 2 x + 4 x^2 - 3 x^3 + x^4) (1 + 3 x + 4 x^2 + 2 x^3 + x^4))/(1 - 36 x^5 - x^10), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 10 2013 *)
LinearRecurrence[{0, 0, 0, 0, 36, 0, 0, 0, 0, 1}, {1, 1, 2, 3, 5, 33, 38, 71, 109, 180}, 40] (* Harvey P. Dale, Sep 30 2016 *)

Prog

(Magma) I:=[1, 1, 2, 3, 5, 33, 38, 71, 109, 180]; [n le 10 select I[n] else 36*Self(n-5)+Self(n-10): n in [1..50]]; // Vincenzo Librandi, Dec 10 2013

Crossrefs

Cf. A010122 (continued fraction for sqrt(13)), A041018 (numerators).

Cf. A041047, A041091, A041151, A041227, A041319, A041427 and A041551. - Johannes W. Meijer, Jun 12 2010

Sequence in context: A109845 A241722 A276043 * A041977 A362640 A261130

Adjacent sequences: A041016 A041017 A041018 * A041020 A041021 A041022

Keyword

nonn,cofr,easy

Author

N. J. A. Sloane

Extensions

More terms from Vincenzo Librandi, Dec 10 2013

Status

approved