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A041042 Numerators of continued fraction convergents to sqrt(27). 2
5, 26, 265, 1351, 13775, 70226, 716035, 3650401, 37220045, 189750626, 1934726305, 9863382151, 100568547815, 512706121226, 5227629760075, 26650854921601, 271736178976085, 1385331749802026, 14125053676996345 (list; graph; refs; listen; history; text; internal format)
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: A277489 A094652 A027529 * A198042 A198157 A048690

Adjacent sequences:  A041039 A041040 A041041 * A041043 A041044 A041045

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 17 16:12 EDT 2022. Contains 353747 sequences. (Running on oeis4.)