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

Author

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

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,9).

Crossrefs

Cf. A002534.

Programs

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

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 *)

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

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

Extensions

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