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%I #8 Sep 08 2022 08:45:03
%S 1,9,39,120,315,753,1687,3612,7470,15040,29634,57366,109421,206115,
%T 384105,709152,1298613,2360943,4264835,7659870,13686456,24340184,
%U 43102644,76031100,133636825,234116493,408900987
%N Second convolution of Lucas numbers A000032(n+1), n >= 0.
%H G. C. Greubel, <a href="/A060929/b060929.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-5,0,3,1).
%F G.f.: ((1+2*x)/(1-x-x^2))^3.
%F a(n) = A060922(n+2, 2) (third column of Lucas triangle).
%F a(n) = (n+1)*((5*n+4)*L(n+2) + (5*n-2)*L(n+1))/10, n >= 1, with the Lucas numbers L(n)=A000032(n)=A000204(n), n >= 1.
%t CoefficientList[Series[((1 + 2*x)/(1 - x - x^2))^3, {x, 0, 50}], x] (* or *) LinearRecurrence[{3,0,-5,0,3,1}, {1,9,39,120,315,753}, 30] (* _G. C. Greubel_, Dec 21 2017 *)
%o (PARI) x='x+O('x^30); Vec(((1+2*x)/(1-x-x^2))^3) \\ _G. C. Greubel_, Dec 21 2017
%o (Magma) I:=[1,9,39,120,315,753]; [n le 6 select I[n] else 3*Self(n-1) - 5*Self(n-3) + 3*Self(n-5) + Self(n-6): n in [1..30]]; // _G. C. Greubel_, Dec 21 2017
%Y Cf. A004799, A060922.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Apr 20 2001
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