A136642 A self-similar scaled version of a Devil's staircase as a triangular sequence.

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

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 structure.

Links

Table of n, a(n) for n=1..88.

Per Bak, Commensurate phases, incommensurate phases and the devil's staircase, Rep. Prog. Phys. 45 (1982) pp.587-629.

E. W. Weisstein, Devils Staircase, MathWorld.

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: A109004 A103823 A355246 * A080382 A349203 A106394

Adjacent sequences: A136639 A136640 A136641 * A136643 A136644 A136645

Keyword

nonn,uned,tabf

Author

Roger L. Bagula, Apr 01 2008

Status

approved