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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 structure.
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LINKS
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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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KEYWORD
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nonn,uned,tabf
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AUTHOR
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STATUS
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approved
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