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A133125 a(n) = (7*3^n - (-3)^n)/6.

%I A133125 #31 Jun 02 2025 04:55:28
%S A133125 1,4,9,36,81,324,729,2916,6561,26244,59049,236196,531441,2125764,
%T A133125 4782969,19131876,43046721,172186884,387420489,1549681956,3486784401,
%U A133125 13947137604,31381059609,125524238436,282429536481,1129718145924,2541865828329,10167463313316
%N A133125 a(n) =  (7*3^n - (-3)^n)/6.
%C A133125 A133647 is a companion sequence.
%H A133125 Andrew Howroyd, <a href="/A133125/b133125.txt">Table of n, a(n) for n = 0..1000</a>
%H A133125 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,9).
%F A133125 A133080 * A000244, where A000244 = (3^0, 3^1, 3^2, ...).
%F A133125 For even n, a(n) = 3^n. For odd n, a(n) = 4 * 3^(n-1).
%F A133125 From _R. J. Mathar_, Oct 30 2008: (Start)
%F A133125 G.f.: (1+4*x)/((1+3*x)*(1-3*x)).
%F A133125 a(n) = 9*a(n-2). (End)
%F A133125 a(n) = A038754(n)^2. - _T. D. Noe_, Jun 10 2011
%F A133125 E.g.f.: (3*exp(3*x) + sinh(3*x))/3. - _Andrew Howroyd_, Jul 03 2024
%F A133125 From _Amiram Eldar_, Jun 02 2025: (Start)
%F A133125 Sum_{n>=0} 1/a(n) = 45/32.
%F A133125 Sum_{n>=0} (-1)^n/a(n) = 27/32. (End)
%e A133125 a(4) = 3^4 = 81.
%e A133125 a(5) = 324 = 4 * 3^4.
%t A133125 Table[3^(n - 2) ((-1)^n + 7)/2, {n, 1, 60}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 10 2011 *)
%t A133125 LinearRecurrence[{0,9},{1,4},30] (* _Harvey P. Dale_, Mar 27 2017 *)
%o A133125 (PARI) a(n) =  (7*3^n - (-3)^n)/6 \\ _Andrew Howroyd_, Jul 03 2024
%Y A133125 Cf. A000244, A038754, A133080, A133647.
%K A133125 nonn,easy
%O A133125 0,2
%A A133125 _Gary W. Adamson_, Sep 19 2007
%E A133125 Name changed by _Andrew Howroyd_, Jul 03 2024