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A152268
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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.
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1
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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; internal format)
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OFFSET
| 0,3
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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
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients, signature (7,-8).
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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.
a(n)=7*a(n-1)-8*a(n-2). G.f.: x/(1-7x+8x^2). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 04 2008]
a(n)=(1/17)*sqrt(17)*{[(7/2)+(1/2)*sqrt(17)]^n-[(7/2)-(1/2)*sqrt(17)]^n}, with n>=0 [From Paolo P. Lava (paoloplava(AT)gmail.com), Feb 11 2009]
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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}]
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PROG
| (Other) sage: [lucas_number1(n, 7, 8) for n in xrange(0, 22)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
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CROSSREFS
| A006131
Sequence in context: A081625 A144635 A097165 * A026002 A173409 A057009
Adjacent sequences: A152265 A152266 A152267 * A152269 A152270 A152271
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KEYWORD
| nonn,easy
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 01 2008
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