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a(n) = ((1 + 3*sqrt(2))^n - (1 - 3*sqrt(2))^n)/(2*sqrt(2)).
2

%I #21 Sep 08 2022 08:45:28

%S 0,3,6,63,228,1527,6930,39819,197448,1071819,5500254,29221431,

%T 151947180,800658687,4184419434,21980036547,115095203472,603851028243,

%U 3164320515510,16594108511151,86981665785972,456063176261511,2390814670884546,12534703338214779

%N a(n) = ((1 + 3*sqrt(2))^n - (1 - 3*sqrt(2))^n)/(2*sqrt(2)).

%H G. C. Greubel, <a href="/A125817/b125817.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from T. D. Noe)

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2, 17).

%F From _Philippe Deléham_, Dec 12 2006: (Start)

%F a(n) = 2*a(n-1) + 17*a(n-2), with a(0)=0, a(1)=3.

%F G.f.: 3*x/(1-2*x-17*x^2). (End)

%t Expand[Table[((1+3Sqrt[2])^n -(1-3Sqrt[2])^n)/(2Sqrt[2]), {n, 0, 30}]]

%t LinearRecurrence[{2, 17}, {0, 3}, 30] (* _T. D. Noe_, Mar 28 2012 *)

%o (PARI) my(x='x+O('x^30)); concat([0], Vec(3*x/(1-2*x-17*x^2))) \\ _G. C. Greubel_, Aug 02 2019

%o (Magma) I:=[0,3]; [n le 2 select I[n] else 2*Self(n-1) +17*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Aug 02 2019

%o (Sage) (3*x/(1-2*x-17*x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 02 2019

%o (GAP) a:=[0,3];; for n in [3..30] do a[n]:=2*a[n-1]+17*a[n-2]; od; a; # _G. C. Greubel_, Aug 02 2019

%K nonn

%O 0,2

%A _Artur Jasinski_, Dec 10 2006