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A213046 Convolution of Lucas numbers and positive integers repeated (A000032 and A008619).

%I A213046 #19 Aug 31 2023 16:38:02
%S A213046 2,3,8,13,25,41,71,116,193,314,514,834,1356,2197,3562,5767,9339,15115,
%T A213046 24465,39590,64067,103668,167748,271428,439190,710631,1149836,1860481,
%U A213046 3010333,4870829,7881179,12752024,20633221,33385262,54018502,87403782,141422304
%N A213046 Convolution of Lucas numbers and positive integers repeated (A000032 and A008619).
%H A213046 Colin Barker, <a href="/A213046/b213046.txt">Table of n, a(n) for n = 0..1000</a>
%H A213046 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-3,0,1).
%F A213046 a(n) = 2*a(n-1) + a(n-2) - 3*a(n-3) + a(n-5).
%F A213046 G.f.: (-2 + x)/((-1 + x)^2*(-1 + 2*x^2 + x^3)).
%F A213046 a(n) = (-9/4 + (3*(-1)^n)/4 + (2^(-n)*((1-t)^n*(-5+2*t) + (1+t)^n*(5+2*t)))/t + (-1-n)/2) where t=sqrt(5). - _Colin Barker_, Feb 09 2017
%t A213046 f[x_] := (1 + x) (1 - x)^2; g[x] := 1 - x - x^2;
%t A213046 s = Normal[Series[(2 - x)/(f[x] g[x]), {x, 0, 60}]]
%t A213046 CoefficientList[s, x]  (* A213046 *)
%t A213046 LinearRecurrence[{2,1,-3,0,1},{2,3,8,13,25},40] (* _Harvey P. Dale_, Aug 31 2023 *)
%o A213046 (Magma) /* By definition */ A008619:=func<n | 1+Floor(n/2)>; [&+[A008619(i)*Lucas(n-i): i in [0..n]]: n in [0..34]];
%o A213046 (PARI) a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,0,-3,1,2]^n*[2;3;8;13;25])[1,1] \\ _Charles R Greathouse IV_, Jan 29 2016
%o A213046 (PARI) Vec((-2 + x)/((-1 + x)^2*(-1 + 2*x^2 + x^3)) + O(x^60)) \\ _Colin Barker_, Feb 09 2017
%Y A213046 Cf. A213500.
%K A213046 nonn,easy
%O A213046 0,1
%A A213046 _Clark Kimberling_, Jun 10 2012