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A179237 a(0) = 1, a(1) = 2; a(n+1) =  6*a(n) + a(n-1) for n>1. 4

%I

%S 1,2,13,80,493,3038,18721,115364,710905,4380794,26995669,166354808,

%T 1025124517,6317101910,38927735977,239883517772,1478228842609,

%U 9109256573426,56133768283165,345911866272416,2131604965917661,13135541661778382,80944854936587953

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

%C a(n)/a(n-1) converges to 1/(sqrt(10) - 3) = 6.16227766017... = A176398.

%H Colin Barker, <a href="/A179237/b179237.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 (6,1).

%F Let M = the 2x2 matrix [2,3; 3,4]. a(n) = term (1,1) in M^n.

%F G.f.: ( -1+4*x ) / ( -1+6*x+x^2 ). a(n) = A005668(n) + A015451(n). - _R. J. Mathar_, Jul 06 2012

%F a(n) = ((3-sqrt(10))^n*(1+sqrt(10))+(-1+sqrt(10))*(3+sqrt(10))^n)/(2*sqrt(10)). - _Colin Barker_, Oct 13 2015

%e a(5) = 3038 = 6*a(5) + a(4) = 6*493 + 80.

%e a(5) = term (1,1) in M^5 where M^5 = [3038, 4215, 4215, 5848].

%t CoefficientList[Series[(-1 + 4 x)/(-1 + 6 x + x^2), {x, 0, 33}], x] (* _Vincenzo Librandi_, Oct 13 2015 *)

%o (PARI) Vec((-1+4*x)/(-1+6*x+x^2) + O(x^40)) \\ _Colin Barker_, Oct 13 2015

%o (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 nonn,easy

%O 0,2

%A _Gary W. Adamson_, Jul 04 2010

%E Corrected by _R. J. Mathar_, Jul 06 2012

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Last modified December 14 09:41 EST 2019. Contains 329979 sequences. (Running on oeis4.)