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A179237 a(0) = 1, a(1) = 2; a(n+1) = 6*a(n) + a(n-1) for n>1.

%I A179237 #17 Feb 14 2024 10:09:32
%S A179237 1,2,13,80,493,3038,18721,115364,710905,4380794,26995669,166354808,
%T A179237 1025124517,6317101910,38927735977,239883517772,1478228842609,
%U A179237 9109256573426,56133768283165,345911866272416,2131604965917661,13135541661778382,80944854936587953
%N A179237 a(0) = 1, a(1) = 2; a(n+1) =  6*a(n) + a(n-1) for n>1.
%C A179237 a(n)/a(n-1) converges to 1/(sqrt(10) - 3) = 6.16227766017... = A176398.
%H A179237 Colin Barker, <a href="/A179237/b179237.txt">Table of n, a(n) for n = 0..1000</a>
%H A179237 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,1).
%F A179237 Let M = the 2x2 matrix [2,3; 3,4]. a(n) = term (1,1) in M^n.
%F A179237 G.f.: ( -1+4*x ) / ( -1+6*x+x^2 ). a(n) = A005668(n) + A015451(n). - _R. J. Mathar_, Jul 06 2012
%F A179237 a(n) = ((3-sqrt(10))^n*(1+sqrt(10))+(-1+sqrt(10))*(3+sqrt(10))^n)/(2*sqrt(10)). - _Colin Barker_, Oct 13 2015
%F A179237 a(n) = Sum_{k=0..n-2} A168561(n-2,k)*6^k + 2 * Sum_{k=0..n-1} A168561(n-1,k)*6^k, n>0. - _R. J. Mathar_, Feb 14 2024
%F A179237 a(n) = A005668(n+1) - 4*A005668(n). - _R. J. Mathar_, Feb 14 2024
%e A179237 a(5) = 3038 = 6*a(5) + a(4) = 6*493 + 80.
%e A179237 a(5) = term (1,1) in M^5 where M^5 = [3038, 4215, 4215, 5848].
%t A179237 CoefficientList[Series[(-1 + 4 x)/(-1 + 6 x + x^2), {x, 0, 33}], x] (* _Vincenzo Librandi_, Oct 13 2015 *)
%o A179237 (PARI) Vec((-1+4*x)/(-1+6*x+x^2) + O(x^40)) \\ _Colin Barker_, Oct 13 2015
%o A179237 (Magma) I:=[1,2]; [n le 2 select I[n] else 6*Self(n-1)+Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Oct 13 2015
%K A179237 nonn,easy
%O A179237 0,2
%A A179237 _Gary W. Adamson_, Jul 04 2010
%E A179237 Corrected by _R. J. Mathar_, Jul 06 2012