A041115 Denominators of continued fraction convergents to sqrt(66).

1, 8, 129, 1040, 16769, 135192, 2179841, 17573920, 283362561, 2284474408, 36834953089, 296964099120, 4788260539009, 38603048411192, 622437035118081, 5018099329355840, 80912026304811521, 652314309767848008, 10517940982590379649

Offset

0,2

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..250

Index entries for linear recurrences with constant coefficients, signature (0,130,0,-1).

Formula

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

From Colin Barker, Feb 28 2013: (Start)

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

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

a(2n) = A041495(2n), a(2n+1) = A041495(2n+1)*2. - M. F. Hasler, Feb 23 2020

Maple

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

Mathematica

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[66], n]]], {n, 1, 60}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
CoefficientList[Series[(1 + 8 x - x^2)/(x^4 - 130 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)

Prog

(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

Crossrefs

Cf. A041114, A041495.

Sequence in context: A104997 A265097 A027951 * A348546 A041112 A348207

Adjacent sequences: A041112 A041113 A041114 * A041116 A041117 A041118

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Status

approved