A041042 Numerators of continued fraction convergents to sqrt(27).

5, 26, 265, 1351, 13775, 70226, 716035, 3650401, 37220045, 189750626, 1934726305, 9863382151, 100568547815, 512706121226, 5227629760075, 26650854921601, 271736178976085, 1385331749802026, 14125053676996345

Offset

0,1

Comments

Subset of |A002316| (conjectured).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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

Formula

G.f.: (-x^3+5x^2+26x+5)/(x^4-52x^2+1).

From Gerry Martens, Jul 11 2015: (Start)

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

a0(n) = ((-5-3*sqrt(3))/(26+15*sqrt(3))^n+(-5+3*sqrt(3))*(26+15*sqrt(3))^n)/2.

a1(n) = (1/(26+15*sqrt(3))^n+(26+15*sqrt(3))^n)/2. (End)

Mathematica

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[27], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011*)
Numerator/@Convergents[Sqrt[27], 20] (* Harvey P. Dale, Jul 21 2011 *)
CoefficientList[Series[(- x^3 + 5 x^2 + 26 x + 5)/(x^4 - 52 x^2 + 1), {x, 0, 30}], x]  (* Vincenzo Librandi, Oct 28 2013 *)
a0[n_] := (-5-3*Sqrt[3]+(-5+3*Sqrt[3])*(26+15*Sqrt[3])^(2*n))/(2*(26+15*Sqrt[3])^n) // Simplify
a1[n_] := (1+(26+15*Sqrt[3])^(2*n))/(2*(26+15*Sqrt[3])^n) //  Simplify
Flatten[MapIndexed[{a0[#], a1[#]}&, Range[10]]] (* Gerry Martens, Jul 10 2015 *)
LinearRecurrence[{0, 52, 0, -1}, {5, 26, 265, 1351}, 30] (* Harvey P. Dale, Dec 12 2015 *)

Crossrefs

Cf. A010482, A041043, A002316.

Sequence in context: A027529 A368294 A366700 * A198042 A198157 A048690

Adjacent sequences: A041039 A041040 A041041 * A041043 A041044 A041045

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane

Status

approved