A041151 - OEIS

%I #26 Sep 08 2022 08:44:54

%S 1,4,5,9,41,747,3029,3776,6805,30996,564733,2289928,2854661,5144589,

%T 23433017,426938895,1731188597,2158127492,3889316089,17715391848,

%U 322766369353,1308780869260,1631547238613,2940328107873,13392859670105,244011802169763,989440068349157

%N Denominators of continued fraction convergents to sqrt(85).

%C From _Johannes W. Meijer_, Jun 12 2010: (Start)

%C The a(n) terms of this sequence can be constructed with the terms of sequence A099371.

%C For the terms of the periodic sequence of the continued fraction for sqrt(85) see A010158. We observe that its period is five. The decimal expansion of sqrt(85) is A010536. (End)

%H Vincenzo Librandi, <a href="/A041151/b041151.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,756,0,0,0,0,1).

%F From _Johannes W. Meijer_, Jun 12 2010: (Start)

%F a(5*n) = A099371(3*n+1), a(5*n+1) = (A099371(3*n+2)-A099371(3*n+1))/2, a(5*n+2) = (A099371(3*n+2)+A099371(3*n+1))/2, a(5*n+3):= A099371(3*n+2) and a(5*n+4) = A099371(3*n+3)/2. (End)

%F G.f.: -(x^8-4*x^7+5*x^6-9*x^5+41*x^4+9*x^3+5*x^2+4*x+1) / (x^10+756*x^5-1). - _Colin Barker_, Nov 11 2013

%F a(n) = 756*a(n-5) + a(n-10). - _Vincenzo Librandi_, Dec 12 2013

%t Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[85], n]]], {n, 1, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 23 2011 *)

%t Denominator[Convergents[Sqrt[85], 30]] (* _Vincenzo Librandi_, Dec 12 2013 *)

%o (Magma) I:=[1, 4, 5, 9, 41, 747, 3029, 3776, 6805, 30996]; [n le 10 select I[n] else 756*Self(n-5)+Self(n-10): n in [1..30]]; // _Vincenzo Librandi_, Dec 12 2013

%Y Cf. A041150, A041019, A041047, A041091, A041151, A041227, A041319, A041427, A041551.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_.