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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
 0, 1, 7, 41, 231, 1289, 7175, 39913, 221991, 1234633, 6866503, 38188457, 212387175, 1181202569, 6569320583, 36535623529, 203194800039, 1130078612041, 6284991883975, 34954314291497, 194400264968679, 1081167340448777 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. LINKS Index entries for linear recurrences with constant coefficients, signature (7,-8). FORMULA M0 = {{0, 1}, {1, 1/2}}; M = {{0, 2}, {2, 1}}; as Mh=M0.M.(M0+I); v[(n)=Mh.v(n-1): a(n) first element of -v(n)[[1]]/2. From R. J. Mathar, Dec 04 2008: (Start) a(n) = 7*a(n-1) - 8*a(n-2). G.f.: x/(1-7x+8x^2). (End) 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 MATHEMATICA Clear[M, M0, Mh, v]; M0 = {{0, 1}, {1, 1/2}}; M = {{0, 2}, {2, 1}}; Mh = M0.(M.Inverse[IdentityMatrix[2] + M0]); v[0] = {0, 1}; v[n_] := v[n] = Mh.v[n - 1] Table[ -v[n][[1]]/2, {n, 0, 30}] PROG (Sage) [lucas_number1(n, 7, 8) for n in range(0, 22)] # Zerinvary Lajos, Apr 23 2009 CROSSREFS Cf. A006131. Sequence in context: A081625 A144635 A097165 * A026002 A173409 A057009 Adjacent sequences:  A152265 A152266 A152267 * A152269 A152270 A152271 KEYWORD nonn,easy AUTHOR Roger L. Bagula, Dec 01 2008 STATUS approved

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Last modified July 24 05:05 EDT 2021. Contains 346273 sequences. (Running on oeis4.)