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a(n) = Sum_{i=0..n} i*L(i), where L = A000032.
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%I #16 Aug 19 2022 15:01:49

%S 0,1,7,19,47,102,210,413,789,1473,2703,4892,8756,15529,27331,47791,

%T 83103,143810,247814,425445,727985,1241981,2113247,3587064,6075432,

%U 10269457,17326975,29185483,49083599,82429278,138244218,231565037

%N a(n) = Sum_{i=0..n} i*L(i), where L = A000032.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,1,1).

%F O.g.f.: x*(-1 - 4*x + x^2)/((-1 + x)*(-1 + x + x^2)^2).

%F a(n) = n*L(n+2) - L(n+3) + 4.

%F a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + a(n-4) + a(n-5). - _Wesley Ivan Hurt_, Aug 19 2022

%t l[n_] := l[n] = l[n - 1] + l[n - 2]; l[0] = 2; l[1] = 1; Table[Sum[k l[k], {k, 0, n}], {n, 0, 10}]

%t With[{nn=40},Accumulate[Times@@@Thread[{LucasL[Range[0,nn]], Range[ 0,nn]}]]] (* _Harvey P. Dale_, Aug 15 2015 *)

%Y Cf. A000032.

%K nonn,easy

%O 0,3

%A Mario Catalani (mario.catalani(AT)unito.it), Jul 20 2004