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

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

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).

Crossrefs

Cf. A010482, A041043, A002316.

Programs

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 *)

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)