A041026 Numerators of continued fraction convergents to sqrt(18).

4, 17, 140, 577, 4756, 19601, 161564, 665857, 5488420, 22619537, 186444716, 768398401, 6333631924, 26102926097, 215157040700, 886731088897, 7309005751876, 30122754096401, 248291038523084

Offset

0,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Index entries for linear recurrences with constant coefficients, signature (0,34,0,-1).

Formula

G.f.: (4+17*x+4*x^2-x^3)/(1-34*x^2+x^4). - Colin Barker, Jan 02 2012

From Gerry Martens, Jul 11 2015: (Start)

Interspersion of 2 sequences [a0(n),a1(n)] for n>0:

a0(n) = ((-4-3*sqrt(2))/(17+12*sqrt(2))^n+(-4+3*sqrt(2))*(17+12*sqrt(2))^n)/2.

a1(n) = (1/(17+12*sqrt(2))^n+(17+12*sqrt(2))^n)/2. (End)

Mathematica

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[18], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
Numerator[Convergents[Sqrt[18], 20]] (* or *) LinearRecurrence[{0, 34, 0, -1}, {4, 17, 140, 577}, 20] (* Harvey P. Dale, Jun 12 2014 *)
a0[n_] := ((-4-3*Sqrt[2])/(17+12*Sqrt[2])^n+(-4+3*Sqrt[2])*(17+12*Sqrt[2])^n)/2 // Simplify
a1[n_] := (1/(17+12*Sqrt[2])^n+(17+12*Sqrt[2])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &, Range[20]]] (* Gerry Martens, Jul 11 2015 *)

Crossrefs

Cf. A010474, A041027.

Sequence in context: A032335 A208803 A156076 * A072755 A320444 A129436

Adjacent sequences: A041023 A041024 A041025 * A041027 A041028 A041029

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane

Status

approved