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A127357 Expansion of 1/(1 - 2*x + 9*x^2). 8
1, 2, -5, -28, -11, 230, 559, -952, -6935, -5302, 51811, 151340, -163619, -1689298, -1906025, 11391632, 39937489, -22649710, -404736821, -605626252, 2431378885, 10313394038, -1255621889 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Hankel transform of A100193. A member of the family of sequences with g.f. 1/(1-2*x+r^2*x^2) which are the Hankel transforms of the sequences given by Sum_{k=0..n} binomial(2*n,k)*r^(n-k).

From Peter Bala, Apr 01 2018: (Start)

With offset 1, this is the Lucas sequence U(n,2,9). The companion Lucas sequence V(n,2,9) is 2*A025172(n).

Define a binary operation o on rational numbers by x o y = (x + y)/(1 - 2*x*y). This is a commutative and associative operation with identity 0. Then 2 o 2 o ... o 2 (n terms) = 2*A127357(n-1)/A025172(n). Cf. A088137 and A087455. (End)

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Wikipedia, Lucas sequence

Index entries for linear recurrences with constant coefficients, signature (2,-9)

FORMULA

a(n) = Sum_{k=0..n} binomial(n-k,k)*2^(n-2*k)*(-9)^k.

a(n) = 2*a(n-1) - 9*a(n-2) for n >= 2. - Vincenzo Librandi, Mar 22 2011

a(n) = ((1-2*sqrt(2)*i)^n-(1+2*sqrt(2)*i)^n)*i/(4*sqrt(2)), where i=sqrt(-1). - Bruno Berselli, Jul 01 2011

From Vladimir Reshetnikov, Oct 15 2016: (Start)

a(n) = 3^n*(cos(n*theta) + sin(n*theta)*sqrt(2)/4), theta = arctan(2*sqrt(2)).

E.g.f.: exp(x)*(cos(2*sqrt(2)*x) + sin(2*sqrt(2)*x)*sqrt(2)/4). (End)

a(n) = 2^n*Product_{k=1..n}(1 + 3*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019

MAPLE

c := 2*sqrt(2): g := exp(x)*(sin(c*x)+c*cos(c*x))/c: ser := series(g, x, 32):

seq(n!*coeff(ser, x, n), n=0..22); # Peter Luschny, Oct 19 2016

MATHEMATICA

Join[{a=1, b=2}, Table[c=2*b-9*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011*)

RootReduce@Table[3^n (Cos[n ArcTan[2 Sqrt[2]]] + Sin[n ArcTan[2 Sqrt[2]]] Sqrt[2]/4), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2016 *)

PROG

(Sage) [lucas_number1(n, 2, 9) for n in range(1, 24)] # Zerinvary Lajos, Apr 23 2009

(MAGMA) m:=23; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-2*x+9*x^2))); // Bruno Berselli, Jul 01 2011

(Maxima) makelist(coeff(taylor(1/(1-2*x+9*x^2), x, 0, n), x, n), n, 0, 22); /* Bruno Berselli, Jul 01 2011 */

(PARI) Vec(1/(1-2*x+9*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

(GAP) a:=[1, 2];; for n in [3..25] do a[n]:=2*a[n-1]-9*a[n-2]; od; a; # Muniru A Asiru, Oct 23 2018

CROSSREFS

Variant is A025170.

Cf. A025172, A088137, A087455.

Sequence in context: A208221 A208224 A208227 * A025170 A151775 A286879

Adjacent sequences:  A127354 A127355 A127356 * A127358 A127359 A127360

KEYWORD

sign,easy

AUTHOR

Paul Barry, Jan 11 2007

STATUS

approved

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Last modified July 8 03:38 EDT 2020. Contains 335504 sequences. (Running on oeis4.)