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A127210 a(n) = 3^n*Lucas(n), where Lucas = A000204.

%I A127210 #23 Jun 29 2023 19:53:54
%S A127210 3,27,108,567,2673,13122,63423,308367,1495908,7263027,35252253,
%T A127210 171124002,830642283,4032042867,19571909148,95004113247,461159522073,
%U A127210 2238515585442,10865982454983,52744587633927,256027604996628,1242784103695227,6032600756055333,29282859201423042
%N A127210 a(n) = 3^n*Lucas(n), where Lucas = A000204.
%H A127210 G. C. Greubel, <a href="/A127210/b127210.txt">Table of n, a(n) for n = 1..1445</a>
%H A127210 Ivica Martinjak, <a href="http://arxiv.org/abs/1508.01444">Two Extensions of the Sury's Identity</a>, arXiv:1508.01444 [math.CO], 2015.
%H A127210 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3, 9).
%F A127210 a(n) = Trace of matrix [({3,3},{3,0})^n] = 3^n * Trace of matrix [({1,1},{1,0})^n].
%F A127210 From _R. J. Mathar_, Oct 27 2008: (Start)
%F A127210 a(n) = 3*a(n-1) + 9*a(n-2).
%F A127210 G.f.: 3*x*(1 + 6*x)/(1 - 3*x - 9*x^2).
%F A127210 a(n) = 3*A099012(n) +18*A099012(n-1). (End)
%t A127210 Table[3^n Tr[MatrixPower[{{1, 1}, {1, 0}}, x]], {x, 1, 20}]
%t A127210 Table[3^n LucasL[n], {n, 25}] (* _Vincenzo Librandi_, Aug 07 2015 *)
%o A127210 (PARI) lucas(n) = fibonacci(n-1) + fibonacci(n+1);
%o A127210 vector(30, n, 3^n*lucas(n)) \\ _Michel Marcus_, Aug 07 2015
%o A127210 (Magma) [3^n*Lucas(n): n in [1..30]]; // _Vincenzo Librandi_, Aug 07 2015
%Y A127210 Cf. A000204, A087131, A127211, A127212, A127213, A127214, A127215, A127216.
%K A127210 nonn,easy
%O A127210 1,1
%A A127210 _Artur Jasinski_, Jan 09 2007
%E A127210 More terms from _Michel Marcus_, Aug 07 2015