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%I #23 Sep 08 2022 08:45:58
%S 0,0,2,10,40,144,490,1610,5168,16320,50930,157546,484120,1480080,
%T 4507162,13683050,41439200,125259264,378051170,1139641930,3432176008,
%U 10328516880,31062778570,93374780426,280574458640,842810055360,2531053642322,7599494558890,22813774416760,68478238362384
%N a(n) = (2/3)*Sum_{i=1..n-1} A000129(i)*3^(n-i).
%C Number of ternary words of length n on {0,1,2} containing the subwords 02 or 20. - _Philippe Deléham_, Apr 27 2012
%H G. C. Greubel, <a href="/A193519/b193519.txt">Table of n, a(n) for n = 0..1000</a>
%H Gy. Tasi and F. Mizukami, <a href="http://dx.doi.org/10.1023/A:1019163812482">Quantum algebraic-combinatoric study of the conformational properties of n-alkanes</a>, J. Math. Chemistry, 25, 1999, 55-64 (see Eq. (19)).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,-3).
%F a(n) = 2*A137212(n).
%F G.f.: 2*x^2/((1-3*x)*(1-2*x-x^2)). - _Philippe Deléham_, Apr 27 2012
%F a(n) = 5*a(n-1) - 5*a(n-2) - 3*a(n-3), a(0) = a(1) = 0, a(2) = 2. - _Philippe Deléham_, Apr 27 2012
%F a(n) = (1/2)*(2*3^n - A002203(n+1)). - _G. C. Greubel_, Jan 05 2022
%e a(3) = 10 because among the 3^3 = 27 ternary words of length 3 only 10, namely 002, 020, 021, 022, 102, 120, 200, 201, 202, 220 contain the subwords 02 or 20. - _Philippe Deléham_, Apr 27 2012
%t Table[(2*3^n - LucasL[n+1, 2])/2, {n, 0, 30}] (* _G. C. Greubel_, Jan 05 2022 *)
%o (Magma) [n le 3 select 2*Floor((n-1)/2) else 5*Self(n-1) -5*Self(n-2) -3*Self(n-3): n in [1..31]]; // _G. C. Greubel_, Jan 05 2022
%o (Sage) [(2*3^n - lucas_number2(n+1, 2, -1))/2 for n in (0..30)] # _G. C. Greubel_, Jan 05 2022
%Y Cf. A000129, A002203, A137212.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Jul 29 2011
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