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 A136642 A self-similar scaled version of a Devil's staircase as a triangular sequence. 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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. REFERENCES 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 LINKS FORMULA t(n,m)=Floor[1+2^m*Winding_Number(Omega)]: 0<=omega<=1;in steps of 1/2^m EXAMPLE {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}, MATHEMATICA 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] CROSSREFS Sequence in context: A159335 A109004 A103823 * A080382 A106394 A171712 Adjacent sequences:  A136639 A136640 A136641 * A136643 A136644 A136645 KEYWORD nonn,uned,tabl AUTHOR Roger L. Bagula, Apr 01 2008 STATUS approved

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