[go: up one dir, main page]

login
A041019
Denominators of continued fraction convergents to sqrt(13).
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
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).
Sequence in context: A109845 A241722 A276043 * A041977 A362640 A261130
KEYWORD
nonn,cofr,frac,easy
EXTENSIONS
More terms from Vincenzo Librandi, Dec 10 2013
STATUS
approved