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%I #14 Oct 15 2021 12:20:50
%S 3,8,20,47,109,253,588,1367,3178,7388,17175,39927,92819,215778,501623,
%T 1166132,2710928,6302143,14650705,34058757,79177004,184064203,
%U 427897358,994740672,2312491503,5375890523,12497429235,29052998162,67540026539,157011512528
%N Expansion of g.f. (3 -x +2*x^2)/(1 -3*x +2*x^2 -x^3).
%C Previous name: Transform of A000027 by the T_{1,2} transformation (see link).
%H G. C. Greubel, <a href="/A136305/b136305.txt">Table of n, a(n) for n = 0..1000</a>
%H Richard Choulet, <a href="https://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 G.f.: f(z) = 3 +8*z + ... = (3 -z +2*z^2)/(1 -3*z +2*z^2 -z^3).
%F a(n+3) = 3*a(n+2) -2*a(n+1) +a(n) (n>=0). - _Richard Choulet_, Apr 07 2009
%t LinearRecurrence[{3,-2,1}, {3,8,20}, 40] (* _G. C. Greubel_, Apr 19 2021 *)
%t CoefficientList[Series[(3-x+2x^2)/(1-3x+2x^2-x^3),{x,0,40}],x] (* _Harvey P. Dale_, Oct 15 2021 *)
%o (Magma) [n le 3 select 2^(n-1)*(n+2) else 3*Self(n-1) - 2*Self(n-2) +Self(n-3): n in [1..41]]; // _G. C. Greubel_, Apr 19 2021
%o (Sage)
%o @CachedFunction
%o def a(n): return 2^n*(n+3) if n<3 else sum((-1)^j*(3-j)*a(n-j-1) for j in (0..2))
%o [a(n) for n in (0..40)] # _G. C. Greubel_, Apr 19 2021
%Y Cf. A097550, A135364, A136302, A136303, A136304, A137229, A137234, A137249.
%K nonn,easy
%O 0,1
%A _Richard Choulet_, Mar 22 2008
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