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%I #49 Dec 22 2023 08:39:32
%S 0,1,3,27,135,891,5103,31347,185895,1121931,6711903,40330467,
%T 241805655,1451365371,8706597903,52244370387,313451873415,
%U 1880754287211,11284396583103,67706766919107,406239439253175,2437440122303451,14624630273467503,87747813021864627,526486783988008935
%N a(n) = 3*a(n-1) + 18*a(n-2), a(0)=0, a(1)=1.
%C The ratio a(n+1)/a(n) converges to 6 as n approaches infinity. - _Felix P. Muga II_, Mar 10 2014
%H Vincenzo Librandi, <a href="/A080424/b080424.txt">Table of n, a(n) for n = 0..1000</a>
%H F. P. Muga II, <a href="https://www.researchgate.net/publication/267327689_Extending_the_Golden_Ratio_and_the_Binet-de_Moivre_Formula">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, March 2014; Preprint on ResearchGate.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,18).
%F G.f.: x/((1+3*x)*(1-6*x)).
%F a(n) = (6^n - (-3)^n)/9.
%F a(n+1) = 6*a(n) + (-3)^n. - _Paul Curtz_, Jun 07 2011
%F a(n) = 3^(n-1)*A001045(n). - _R. J. Mathar_, Mar 08 2021
%t a[n_]:=(6^n - (-3)^n)/9; Array[a, 22, 0] (* _Robert G. Wilson v_, Aug 13 2011 *)
%t LinearRecurrence[{3,18}, {0,1}, 31] (* _G. C. Greubel_, Dec 22 2023 *)
%o (PARI) a(n)=(6^n-(-3)^n)/9 \\ _Charles R Greathouse IV_, Jun 10 2011
%o (Magma) [(6^n-(-3)^n)/9: n in [0..30]]; // _Vincenzo Librandi_, Aug 13 2011
%o (SageMath) [3^(n-1)*lucas_number1(n,1,-2) for n in range(31)] # _G. C. Greubel_, Dec 22 2023
%Y Cf. A001045, A015441, A051958, A079773.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Feb 24 2003
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