A041043 Denominators of continued fraction convergents to sqrt(27).

1, 5, 51, 260, 2651, 13515, 137801, 702520, 7163001, 36517525, 372338251, 1898208780, 19354426051, 98670339035, 1006057816401, 5128959421040, 52295652026801, 266607219555045, 2718367847577251, 13858446457441300

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,52,0,-1).

Formula

a(n) = 52*a(n-2)-a(n-4). G.f.: -(x^2-5*x-1)/(x^4-52*x^2+1). - Colin Barker, Jul 15 2012

From Gerry Martens, Jul 11 2015: (Start)

Interspersion of 2 sequences [a0(n),a1(n)]:

a0(n) = ((9+5*sqrt(3))/(26+15*sqrt(3))^n+(9-5*sqrt(3))*(26+15*sqrt(3))^n)/18.

a1(n) = (-1/(26+15*sqrt(3))^n+(26+15*sqrt(3))^n)/(6*sqrt(3)). (End)

Mathematica

Denominator[Convergents[Sqrt[27], 50]] (* Harvey P. Dale, Apr 22 2012 *)
CoefficientList[Series[- (x^2 - 5 x - 1)/(x^4 - 52 x^2 + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 22 2013 *)
a0[n_] := (9+5*Sqrt[3]+(9-5*Sqrt[3])*(26+15*Sqrt[3])^(2*n))/(18*(26+15*Sqrt[3])^n) // Simplify
a1[n_] := (-1+(26+15*Sqrt[3])^(2*n))/(6*Sqrt[3]*(26+15*Sqrt[3])^n) // FullSimplify
Flatten[MapIndexed[{a0[#], a1[#]}&, Range[10]]] (* Gerry Martens, Jul 10 2015 *)

Crossrefs

Cf. A010482, A041042.

Sequence in context: A134938 A068540 A208997 * A362516 A369295 A145641

Adjacent sequences: A041040 A041041 A041042 * A041044 A041045 A041046

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane

Status

approved