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A136642
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A self-similar scaled version of a Devil's staircase as a triangular sequence.
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0
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1, 1, 2, 1, 2, 3, 1, 1, 3, 4, 5, 1, 1, 2, 3, 5, 6, 7, 8, 9, 1, 1, 1, 2, 4, 5, 6, 8, 9, 9, 11, 12, 13, 15, 16, 16, 17, 1, 1, 1, 1, 1, 1, 4, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 17, 18, 20, 21, 22, 23, 25, 26, 27, 29, 32, 32, 32, 32, 32, 33, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 7, 9, 11, 12, 13, 15
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OFFSET
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1,3
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COMMENTS
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Row sums are:
{1, 3, 6, 14, 42, 146, 546, 2114, 8322, 33026};
This method is a way to get a Sierpinski-type ratio of 2 growth factor
of Self-Similarity into a Cantor-like devil's staircase.
Putting them together gives a new fractal with:
Sort[Flatten[Table[a1[[n]], {n, 1, Length[a1]}]]]
having a different stricture.
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REFERENCES
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Per Bak,1982, "Commensurate phases, incommensurate phases and the devil's staircase", in: Reports on Progress in Physics, Vol 45, pp.587-629.
Weisstein, Eric W. "Devil's Staircase." http://mathworld.wolfram.com/DevilsStaircase.html
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LINKS
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Table of n, a(n) for n=1..88.
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FORMULA
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t(n,m)=Floor[1+2^m*Winding_Number(Omega)]: 0<=omega<=1;in steps of 1/2^m
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EXAMPLE
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{1},
{1, 2},
{1, 2, 3},
{1, 1, 3, 4, 5},
{1, 1, 2, 3, 5, 6, 7, 8, 9},
{1, 1, 1, 2, 4, 5, 6, 8, 9, 9, 11, 12, 13, 15, 16, 16, 17},
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MATHEMATICA
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f[{omega_, t_}] := {omega, t + omega - Sin[2Pi t]/(2Pi)}; WindingNumber[n_, {omega_, t_}] := (Nest[f, {omega, t}, n][[2]] - t)/n; a = Table[Table[Floor[1 + 2^n*WindingNumber[1000, {omega, 0}]], {omega, 0, 1, N[1/2^n]}], {n, 0, 8}]; a1 = Join[{{1}}, a]; Flatten[a1]
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CROSSREFS
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Sequence in context: A159335 A109004 A103823 * A080382 A106394 A171712
Adjacent sequences: A136639 A136640 A136641 * A136643 A136644 A136645
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KEYWORD
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nonn,uned,tabl
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AUTHOR
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Roger L. Bagula, Apr 01 2008
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STATUS
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approved
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