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A152268 A hidden Markov recursion involving the matrices: M0 = {{0, 1}, {1, 1/2}}; M = {{0, 2}, {2, 1}}; as Mh=M0.M.(M0+I); v[(n)=Mh.v(n-1): first element of v. 1

%I

%S 0,1,7,41,231,1289,7175,39913,221991,1234633,6866503,38188457,

%T 212387175,1181202569,6569320583,36535623529,203194800039,

%U 1130078612041,6284991883975,34954314291497,194400264968679,1081167340448777

%N A hidden Markov recursion involving the matrices: M0 = {{0, 1}, {1, 1/2}}; M = {{0, 2}, {2, 1}}; as Mh=M0.M.(M0+I); v[(n)=Mh.v(n-1): first element of v.

%C Characteristic Polynomial is: 8 - 7 x + x^2. Binary switching of the IdentityMatrix[2] uncovers opposite signed A006131 with characteristic polynomial -4 - x + x^2.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-8).

%F M0 = {{0, 1}, {1, 1/2}}; M = {{0, 2}, {2, 1}};

%F as Mh=M0.M.(M0+I); v[(n)=Mh.v(n-1):

%F a(n) first element of -v(n)[[1]]/2.

%F From _R. J. Mathar_, Dec 04 2008: (Start)

%F a(n) = 7*a(n-1) - 8*a(n-2).

%F G.f.: x/(1-7x+8x^2). (End)

%F a(n) = (1/17)*sqrt(17)*((7/2 + (1/2)*sqrt(17))^n - (7/2 - (1/2)*sqrt(17))^n), with n >= 0. - _Paolo P. Lava_, Feb 11 2009

%t Clear[M, M0, Mh, v];

%t M0 = {{0, 1}, {1, 1/2}}; M = {{0, 2}, {2, 1}};

%t Mh = M0.(M.Inverse[IdentityMatrix[2] + M0]);

%t v[0] = {0, 1};

%t v[n_] := v[n] = Mh.v[n - 1]

%t Table[ -v[n][[1]]/2, {n, 0, 30}]

%o (Sage) [lucas_number1(n,7,8) for n in range(0, 22)] # _Zerinvary Lajos_, Apr 23 2009

%Y Cf. A006131.

%K nonn,easy

%O 0,3

%A _Roger L. Bagula_, Dec 01 2008

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Last modified December 4 21:58 EST 2020. Contains 338941 sequences. (Running on oeis4.)