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Search: id:A117648
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| A117648 |
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Simplest bounching ball system at: f[n]=(4/3)*f[n-1]-f[n-2]. |
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| 2, 3, 2, 1, 3, 3, 2, 0, 2, 2, 0, 2, 3, 3, 1, 2, 2, 1, 1, 3, 3, 2, 1, 2, 2, 0, 2, 3, 3, 1, 2, 2, 1, 1, 3, 3, 2, 1, 2, 2, 0, 2, 3, 3, 0, 2, 2, 1, 1, 3, 3
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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There is a large family of this type of linear harmonic equation. General equation form is: (x - (a + I*Sqrt[2*a + 1])/(a + 1))*(x - (a - I*Sqrt[2*a + 1])/(a + 1)) Rational numbers are Coefficients: Table[Coefficient[ExpandAll[(x - (a + I*Sqrt[2*a + 1])/( a + 1))*(x - (a - I*Sqrt[2*a + 1])/(a + 1))], x], {a, 1, 285}] Binet solution for this one is: f[n]= 2*Cos[n*ArcTan[Sqrt[5]/2]]+Sqrt[5]*Sin[n*ArcTan[Sqrt[5]/2]]
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FORMULA
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f[0]=2;f[1]=3; f[n]=(4/3)*f[n-1]-f[n-2] a(n) = Abs[Floor[f[n]]]
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MATHEMATICA
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Clear[f, M, v] f[0] = 2; f[1] = 3; f[n_] := f[n] = (4/3)*f[n - 1] - f[n - 2] Table[Abs[Floor[f[n]]], {n, 0, 50}] ListPlot[%, PlotJoined -> True] M = {{0, 1}, {-1, (4/3)}}; v[0] = {2, 3}; v[n_] := v[n] = M.v[n - 1] Table[Abs[Floor[v[n][[1]]]], {n, 0, 80}] ListPlot[%]
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CROSSREFS
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Sequence in context: A118105 A125211 A139367 this_sequence A037222 A102288 A107357
Adjacent sequences: A117645 A117646 A117647 this_sequence A117649 A117650 A117651
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KEYWORD
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nonn,uned,probation
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 10 2006
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