login
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
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
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
KEYWORD
nonn,uned,tabf
AUTHOR
Roger L. Bagula, Apr 01 2008
STATUS
approved