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A117648 Simplest bounching ball system at: f[n]=(4/3)*f[n-1]-f[n-2]. 0
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; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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]]

LINKS

Table of n, a(n) for n=0..50.

FORMULA

f[0]=2;f[1]=3; f[n]=(4/3)*f[n-1]-f[n-2] a(n) = Abs[Floor[f[n]]]

MATHEMATICA

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[%]

CROSSREFS

Sequence in context: A118105 A125211 A139367 * A037222 A107357 A251718

Adjacent sequences:  A117645 A117646 A117647 * A117649 A117650 A117651

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, Apr 10 2006

STATUS

approved

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Last modified May 11 00:21 EDT 2021. Contains 343784 sequences. (Running on oeis4.)