A127357 Expansion of 1/(1 - 2*x + 9*x^2).

1, 2, -5, -28, -11, 230, 559, -952, -6935, -5302, 51811, 151340, -163619, -1689298, -1906025, 11391632, 39937489, -22649710, -404736821, -605626252, 2431378885, 10313394038, -1255621889

Offset: 0

Author

Paul Barry, Jan 11 2007

Keywords

• sign
• easy

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).

Crossrefs

Variant is A025170.

Cf. A025172, A087455, A088137, A100193, A127357.

Programs

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

Magma

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

Magma

[3^n*Evaluate(ChebyshevU(n+1),1/3): n in [0..50]]; // G. C. Greubel, Jan 02 2024

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

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 *)
CoefficientList[Series[1/(1-2x+9x^2),{x,0,40}],x] (* or *)
LinearRecurrence[ {2,-9},{1,2},40] (* Harvey P. Dale, Mar 15 2022 *)
Table[3^n*ChebyshevU[n, 1/3], {n,0,40}] (* G. C. Greubel, Jan 02 2024 *)

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

Sage

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

SageMath

[3^n*chebyshev_U(n,1/3) for n in range(41)] # G. C. Greubel, Jan 02 2024

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

From G. C. Greubel, Jan 02 2024: (Start)

a(n) = (-1)^n * A025170(n).

a(n) = 3^n * ChebyshevU(n, 1/3). (End)