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A041010 Numerators of continued fraction convergents to sqrt(8). 2
1, 2, 3, 14, 17, 82, 99, 478, 577, 2786, 3363, 16238, 19601, 94642, 114243, 551614, 665857, 3215042, 3880899, 18738638, 22619537, 109216786, 131836323, 636562078, 768398401, 3710155682, 4478554083, 21624372014, 26102926097 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

FORMULA

a(n) = 6*a(n-2) - a(n-4).

a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 4*a(2n) + a(2n-1).

a(2n) = A001333(2n), a(2n+1) = 2*A001333(2n+1).

G.f.: (1+2*x-3*x^2+2*x^3)/(1-6*x^2+x^4).

a(n) = A001333(n)*A000034(n). - R. J. Mathar, Jul 08 2009

From Gerry Martens, Jul 11 2015: (Start)

Interspersion of 2 sequences [a1(n-1),a0(n)] for n>0:

a0(n) = -((3-2*sqrt(2))^n*(1+sqrt(2)))+(-1+sqrt(2))*(3+2*sqrt(2))^n.

a1(n) = ((3-2*sqrt(2))^n+(3+2*sqrt(2))^n)/2. (End)

MATHEMATICA

Join[{1}, Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[8], n]]], {n, 1, 50}]] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)

CoefficientList[Series[(1 + 2 x - 3 x^2 + 2 x^3)/(1 - 6 x^2 + x^4), {x, 0, 30}], x]  (* Vincenzo Librandi_, Oct 28 2013 *)

a0[n_] := -((3-2*Sqrt[2])^n*(1+Sqrt[2]))+(-1+Sqrt[2])*(3+2*Sqrt[2])^n // Simplify

a1[n_] := ((3-2*Sqrt[2])^n+(3+2*Sqrt[2])^n)/2 // Simplify

Flatten[MapIndexed[{a1[#-1], a0[#]} &, Range[20]]] (* Gerry Martens, Jul 11 2015 *)

CROSSREFS

Cf. A010466, A041011.

Sequence in context: A042367 A100341 A041869 * A041733 A212112 A107083

Adjacent sequences:  A041007 A041008 A041009 * A041011 A041012 A041013

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Entry improved by Michael Somos

STATUS

approved

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Last modified August 20 20:09 EDT 2017. Contains 290837 sequences.