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

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

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).

Crossrefs

Cf. A010482, A041042.

Programs

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 *)

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)