%I #32 Sep 08 2022 08:44:54

%S 1,8,129,1040,16769,135192,2179841,17573920,283362561,2284474408,

%T 36834953089,296964099120,4788260539009,38603048411192,

%U 622437035118081,5018099329355840,80912026304811521,652314309767848008,10517940982590379649

%N Denominators of continued fraction convergents to sqrt(66).

%H Nathaniel Johnston, <a href="/A041115/b041115.txt">Table of n, a(n) for n = 0..250</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,130,0,-1).

%F a(n) = 16*a(n-1) + a(n-2) for n >= 2 even and a(n) = 8*a(n-1) + a(n-2) for n >= 2 odd. - _Nathaniel Johnston_, Jun 26 2011

%F From _Colin Barker_, Feb 28 2013: (Start)

%F a(n) = 130*a(n-2) - a(n-4).

%F G.f.: -(x^2 - 8*x - 1) / (x^4 - 130*x^2 + 1). (End)

%F a(2n) = A041495(2n), a(2n+1) = A041495(2n+1)*2. - _M. F. Hasler_, Feb 23 2020

%p a := proc(n) option remember: if(n<=1)then return (n+1)^3: fi: if(n mod 2 = 0)then return 16*a(n-1) + a(n-2): else return 8*a(n-1) + a(n-2): fi: end: seq(a(n),n=0..20); # _Nathaniel Johnston_, Jun 26 2011

%t Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[66],n]]],{n,1,60}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 26 2011 *)

%t CoefficientList[Series[(1 + 8 x - x^2)/(x^4 - 130 x^2 + 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Dec 11 2013 *)

%o (Magma) I:=[1,8,129,1040]; [n le 4 select I[n] else 130*Self(n-2)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Dec 11 2013

%Y Cf. A041114, A041495.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_