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Search: id:A114775
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| A114775 |
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Matrix Markov of a complex sequence based on the Minimal Pisot (A000931) characteristic cubic having its roots rotated by 90 degrees or "I". |
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+0
1
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| 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 3, 4, 5, 7, 5, 7, 9, 12, 9, 12, 16, 21, 16, 21, 28, 37, 28, 37, 49, 65, 49, 65, 86, 114, 86, 114, 151, 200, 151, 200, 265, 351, 265
(list; graph; listen)
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OFFSET
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0,14
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COMMENT
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This procedure essentially divides the sequence A000931 into two parts as real and imaginary. As such it is possible to plot the result on the complex plane where it gives a square outward spiral: b = Table[{Re[v[n][[1]]], Im[v[n][[1]]]}, {n, 0, 25}] ListPlot[b, PlotJoined -> True]
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FORMULA
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M = {{0, 1, 0}, {0, 0, 1}, {-I, -1, 0}}; v[0] = {0, 1, 1}; v[n_] := v[n] = M.v[n - 1] {a(n),a(n+1)}= {Abs[Re[v[n][[1]]]], Abs[Im[v[n][[1]]]]}
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EXAMPLE
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Derivation:
a[n_] := x /. Solve[x^3 - x - 1 == 0, x][[n]]
FullSimplify[ExpandAll[Product[(x - I*a[n]), {n, 1, 3}]]]
Gaves the polynomial:
x^3+x+I
as does:
Det[M-x*IdentityMatrix[3]]
NSolve[Det[M-x*IdentityMatrix[3]]\[Equal]0,x]
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MATHEMATICA
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M = {{0, 1, 0}, {0, 0, 1}, {-I, -1, 0}}; v[0] = {0, 1, 1}; v[n_] := v[n] = M.v[n - 1] aout = Flatten[Table[{Abs[Re[v[n][[1]]]], Abs[Im[v[n][[1]]]]}, {n, 0, 25}]]
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CROSSREFS
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Cf. A000931.
Sequence in context: A164978 A119789 A025424 this_sequence A071136 A025425 A085501
Adjacent sequences: A114772 A114773 A114774 this_sequence A114776 A114777 A114778
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KEYWORD
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nonn,uned
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AUTHOR
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Roger Bagula (rlbagulatftn(AT)yahoo.com), Feb 21 2006
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