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%I #19 Nov 27 2023 06:18:54
%S 1,2,8,30,108,378,1296,4374,14580,48114,157464,511758,1653372,5314410,
%T 17006112,54206982,172186884,545258466,1721868840,5423886846,
%U 17046501516,53464027482,167365651248,523017660150,1631815099668,5083731656658,15816054042936
%N a(n) = Sum_{k=0..3^(n-1)} gcd(k,3^(n-1)) for n > 0 and a(0) = 1.
%H G. C. Greubel, <a href="/A199923/b199923.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9).
%F a(n) = 2 * (n+2) * 3^(n-2), n > 0. - _Sean A. Irvine_, Jun 27 2022
%F From _G. C. Greubel_, Nov 24 2023: (Start)
%F a(n) = (2*(n+2)*3^n + 5*[n=0])/9.
%F G.f.: (9 - 6*x))/(9*(1 - 3*x)^2).
%F E.g.f.: (1/9)*( 5 + 2*(2 + 3*x)*exp(3*x) ). (End)
%e a(3) = 9 + 1 + 1 + 3 + 1 + 1 + 3 + 1 + 1 + 9.
%t LinearRecurrence[{6,-9}, {1,2,8}, 41] (* _G. C. Greubel_, Nov 24 2023 *)
%o (Magma) [1] cat [n le 2 select 2^(2*n-1) else 6*Self(n-1) -9*Self(n-2): n in [1..40]]; // _G. C. Greubel_, Nov 24 2023
%o (SageMath) [(2*(n+2)*3^n + 5*int(n==0))//9 for n in range(41)] # _G. C. Greubel_, Nov 24 2023
%Y First differences of A086972.
%Y Row sums of A199922.
%K nonn
%O 0,2
%A _Peter Luschny_, Nov 12 2011
%E More terms from _Sean A. Irvine_, Jun 27 2022
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