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A086972 a(n) = n*3^(n-1) + (3^n + 1)/2.

%I A086972 #22 Nov 24 2023 12:42:41
%S A086972 1,3,11,41,149,527,1823,6197,20777,68891,226355,738113,2391485,
%T A086972 7705895,24712007,78918989,251105873,796364339,2518233179,7942120025,
%U A086972 24988621541,78452649023,245818300271,768835960421,2400651060089
%N A086972 a(n) = n*3^(n-1) + (3^n + 1)/2.
%C A086972 Binomial transform of A057711 (without leading zero). Second binomial transform of (1,1,3,3,5,5,7,7,9,9,11,11,...).
%H A086972 Vincenzo Librandi, <a href="/A086972/b086972.txt">Table of n, a(n) for n = 0..400</a>
%H A086972 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-15,9).
%F A086972 a(n) = (1/2)*(A081038(n) + 1).
%F A086972 G.f.: (1-4*x+5*x^2)/((1-x)*(1-3*x)^2).
%F A086972 a(n) = A027471(n) + A007051(n).
%F A086972 E.g.f.: (1/2)*( exp(x) + (2*x+1)*exp(3*x) ). - _G. C. Greubel_, Nov 24 2023
%t A086972 Table[((2*n+3)*3^(n-1) +1)/2, {n,0,30}] (* _G. C. Greubel_, Nov 24 2023 *)
%o A086972 (Magma) [n*3^(n-1) + (3^n+1)/2: n in [0..30]]; // _Vincenzo Librandi_, Jun 09 2011
%o A086972 (PARI) Vec((1-4*x+5*x^2)/((1-x)*(1-3*x)^2) + O(x^40)) \\ _Michel Marcus_, Mar 08 2016
%o A086972 (SageMath) [((2*n+3)*3^(n-1) +1)//2 for n in range(31)] # _G. C. Greubel_, Nov 24 2023
%Y A086972 Cf. A007051, A027471, A057711, A081038.
%Y A086972 Partial sums of A199923.
%K A086972 easy,nonn
%O A086972 0,2
%A A086972 _Paul Barry_, Jul 26 2003