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A106851
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Let M = {{0, 0, 0, 1}, {1, 4, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 4}}, v[1] = {0, 1, 1, 2}', v[n]=M.v[n-1]; then a(n) = v[n][[1]]
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0
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0, 2, 9, 37, 152, 626, 2585, 10701, 44400, 184610, 769065, 3209461, 13415048, 56153618, 235357241, 987609501, 4148575200, 17443003202, 73402179657, 309116995525, 1302649664888, 5492768393906, 23173154692697, 97810060234605
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| A 4 X 4 vector Markov chain with characteristic polynomial x^4-8*x^3+16*x^2-1..
Real-valued roots: {{x -> -0.236068}, {x -> 0.267949}, {x -> 3.73205}, {x -> 4.23607}}
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FORMULA
| G.f.: (-3*x^3 - 7*x^2 + 2*x)/[(1-4*x-x^2)*(1-4*x+x^2) ].
a(n) = (1/2) * [A001834(n-1) + Fibonacci(3n+1) ]. -- Ralf Stephan, Nov 18 2010
a(0)=0, a(1)=2, a(2)=9, a(3)=37, a(n)=8*a(n-1)-16*a(n-2)+a(n-4) [From Harvey P. Dale, Aug 05 2011]
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MATHEMATICA
| M = {{0, 0, 0, 1}, {1, 4, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 4}} v[1] = {0, 1, 1, 2}; v[n_] := v[n] = M.v[n - 1]; digits = 50; a = Table[v[n][[1]], {n, 1, digits}]
CoefficientList[Series[(-3 x^3-7x^2+2x)/((1-4x-x^2)(1-4x+x^2)), {x, 0, 30}], x] (* or *) LinearRecurrence[{8, -16, 0, 1}, {0, 2, 9, 37}, 31] (* From Harvey P. Dale, Aug 05 2011 *)
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CROSSREFS
| Sequence in context: A037553 A178875 A012493 * A129169 A162548 A150983
Adjacent sequences: A106848 A106849 A106850 * A106852 A106853 A106854
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KEYWORD
| nonn
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 30 2005
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 09 2007
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