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Transform of the finite sequence (1, 0, -1, 0, 1, 0, -1) by the T_{1,1} transform (see link).
2

%I #13 Sep 08 2022 08:45:43

%S 2,4,9,23,55,126,293,680,1581,3676,8546,19867,46185,107367,249598,

%T 580245,1348906,3135826,7289911,16946987,39396965,91586832,212913553,

%U 494963960,1150651606,2674940451,6218482101,14456217007,33606627270

%N Transform of the finite sequence (1, 0, -1, 0, 1, 0, -1) by the T_{1,1} transform (see link).

%H G. C. Greubel, <a href="/A159331/b159331.txt">Table of n, a(n) for n = 0..1000</a>

%H R. Choulet, <a href="http://www.apmep.fr/IMG/pdf/curtz1.pdf">Curtz-like transformation</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,1).

%F O.g.f.: f(z) = ((1-z)^2/(1-3*z+2*z^2-z^3))*(1-z^2+z^4+z^6) + (z/(1-3*z+2*z^2-z^3)) + ((1-z+z^2)/(1-3*z+2*z^2-z^3)).

%F a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) for n >= 9, with a(0)=2, a(1)=4, a(2)=9, a(3)=23, a(4)=55, a(5)=126, a(6)=293, a(7)=680, a(8)=1581.

%t Join[{2, 4, 9, 23, 55, 126}, LinearRecurrence[{3, -2, 1}, {293, 680, 1581}, 45]] (* _G. C. Greubel_, Jun 26 2018 *)

%o (PARI) z='z+O('z^30); Vec(((1-z)^2/(1-3*z+2*z^2-z^3))*(1-z^2+z^4+z^6) + (z/(1-3*z+2*z^2-z^3)) + ((1-z+z^2)/(1-3*z+2*z^2-z^3))) \\ _G. C. Greubel_, Jun 26 2018

%o (Magma) I:=[293, 680, 1581]; [2, 4, 9, 23, 55, 126] cat [n le 3 select I[n] else 3*Self(n-1) - 2*Self(n-2) + Self(n-3): n in [1..30]]; // _G. C. Greubel_, Jun 26 2018

%Y Cf. A159328, A159329, A159330.

%K easy,nonn

%O 0,1

%A _Richard Choulet_, Apr 10 2009