A123007 Expansion of x*(1+x)/(1 -2*x -9*x^2).

1, 3, 15, 57, 249, 1011, 4263, 17625, 73617, 305859, 1274271, 5301273, 22070985, 91853427, 382345719, 1591372281, 6623856033, 27570062595, 114754829487, 477640222329, 1988073910041, 8274909821043, 34442484832455

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

Formula

From Philippe Deléham, Oct 18 2006: (Start)

a(n) = 2*a(n-1) + 9*a(n-2) for n > 2.

G.f.: x*(1+x)/(1 -2*x -9*x^2). (End)

a(n) = A002534(n) + A002534(n-1). - R. J. Mathar, Sep 17 2013

a(n) = (3*i)^(n-2)*(3*i*chebyshev_U(n-1, -i/3) + chebyshev_U(n-2, -i/3)). - G. C. Greubel, Jul 12 2021

Mathematica

M:= {{0, 3}, {3, 2}}; v[1]= {1, 1}; v[n_]:= v[n]= M.v[n-1];
Table[v[n][[1]], {n, 30}]
Rest[CoefficientList[Series[(x(x+1))/(1-2x-9x^2), {x, 0, 30}], x]] (* or *) LinearRecurrence[{2, 9}, {1, 3}, 30] (* Harvey P. Dale, Aug 07 2015 *)

Prog

(Magma) [n le 2 select 3^(n-1) else 2*Self(n-1) +9*Self(n-2): n in [1..30]]; // G. C. Greubel, Jul 12 2021
(Sage) [(3*i)^(n-2)*(3*i*chebyshev_U(n-1, -i/3) + chebyshev_U(n-2, -i/3)) for n in [1..30]] # G. C. Greubel, Jul 12 2021

Crossrefs

Cf. A002534.

Sequence in context: A218657 A218804 A125673 * A121695 A343994 A017949

Adjacent sequences: A123004 A123005 A123006 * A123008 A123009 A123010

Keyword

nonn,easy

Author

Roger L. Bagula and Gary W. Adamson, Sep 23 2006

Extensions

Definition replaced with the Deléham formula by R. J. Mathar, Sep 17 2013

Status

approved