A123007 - OEIS

%I #26 Sep 08 2022 08:45:28

%S 1,3,15,57,249,1011,4263,17625,73617,305859,1274271,5301273,22070985,

%T 91853427,382345719,1591372281,6623856033,27570062595,114754829487,

%U 477640222329,1988073910041,8274909821043,34442484832455

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

%H G. C. Greubel, <a href="/A123007/b123007.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,9).

%F From _Philippe Deléham_, Oct 18 2006: (Start)

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

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

%F a(n) = A002534(n) + A002534(n-1). - _R. J. Mathar_, Sep 17 2013

%F 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

%t M:= {{0, 3}, {3, 2}}; v[1]= {1, 1}; v[n_]:= v[n]= M.v[n-1];

%t Table[v[n][[1]], {n,30}]

%t 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 *)

%o (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

%o (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

%Y Cf. A002534.

%K nonn,easy

%O 1,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 23 2006

%E Definition replaced with the Deléham formula by _R. J. Mathar_, Sep 17 2013