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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

Table of n, a(n) for n=0..21.

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 xrange(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 October 16 02:52 EDT 2019. Contains 328038 sequences. (Running on oeis4.)