A136640 A limited integer Devil's staircase from a winding number function.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 9, 13, 16, 18, 20, 22, 24, 26, 28, 29, 31, 33, 34, 36, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 51, 51, 53, 54, 56, 58, 58, 60, 61, 63, 65, 67, 67, 67, 67, 67, 67, 70, 72, 73, 75, 76, 77, 79

Offset: 1

Author

Roger L. Bagula, Apr 01 2008

Keywords

• nonn
• less
• uned

Comments

Designed to be integer and 200 in length, this function is a limited representation of a Devil's staircase function; a projection of a set of rational numbers onto the integers.

Links

• Per Bak, Commensurate phases, incommensurate phases and the devil's staircase, Rep. Prog. Phys. 45 (1982) pp.587-629.
• Eric Weisstein's World of Mathematics, Devil's Staircase.

Programs

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[Floor[1+200*WindingNumber[1000,{omega,0}]], {omega,0,1,.005}]

Formula

a(n) = floor(1+200*Winding_Number(Omega)): 0<=omega<=1;in steps of 1/200