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%I #35 Jun 25 2023 20:52:07
%S 1,6,17,38,80,158,303,566,1039,1880,3364,5964,10493,18342,31885,55162,
%T 95032,163114,279051,475990,809771,1374316,2327372,3933528,6636025,
%U 11176518,18794633,31560206,52925984,88646390,148303719,247841654
%N Self-convolution of Lucas numbers.
%H Alois P. Heinz, <a href="/A004799/b004799.txt">Table of n, a(n) for n = 1..1000</a>
%H É. Czabarka, R. Flórez, and L. Junes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Florez/florez12.html">A Discrete Convolution on the Generalized Hosoya Triangle</a>, Journal of Integer Sequences, 18 (2015), #15.1.6.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2, 1, -2, -1).
%F From _Wolfdieter Lang_, Apr 24 2001: (Start)
%F a(n) = A060922(n, 1) (second column of Lucas triangle).
%F a(n) = ((-4 + 5*n)*L(n+1) + 2*L(n))/5 with L(n) = A000032(n) = A000204(n), n >= 1.
%F G.f.: x*((1+2*x)/(1-x-x^2))^2. (End)
%p a:= n-> (Matrix([[17, 6, 1, 0]]). Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [2, 1, -2, -1][i] else 0 fi)^n) [1,4]: seq (a(n), n=1..40); # _Alois P. Heinz_, Oct 28 2008
%t a[n_]:= ((5*n-4)*LucasL[n+1] + 2*LucasL[n])/5; Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Nov 12 2015 *)
%o (PARI) Vec(x*((1+2*x)/(1-x-x^2))^2 + O(x^50)) \\ _Altug Alkan_, Nov 12 2015
%o (Magma) [((5*n-4)*Lucas(n+1) + 2*Lucas(n))/5: n in [1..30]]; // _G. C. Greubel_, Dec 17 2017
%o (Sage) [((5*n-4)*lucas_number2(n+1,1,-1) + 2*lucas_number2(n,1,-1))/5 for n in (1..30)] # _G. C. Greubel_, Apr 07 2021
%Y Cf. A000032, A000204, A060922.
%K nonn
%O 1,2
%A _Clark Kimberling_
%E More terms from _Alois P. Heinz_, Oct 28 2008
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