A272532 Single bit representation of the sum of two sinusoids with periods 2 and 2*sqrt(2).

1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1

Offset: 1

Author

Andres Cicuttin, May 02 2016

Keywords

• nonn
• base

Comments

Since the ratio of the two periods is irrational, the sequence is strictly non-periodic.

From the factorized expression of the corresponding real function of x : 2*cos(2Pi((2 - sqrt(2))/8)x)*sin(2Pi((2 + sqrt(2))/8)x), it is possible to see that the largest distance between consecutive zeros is not greater than the shortest semi-period, 4/(2 + sqrt(2)), that is smaller than 2, and from this it follows that there are no more than two consecutive 0's or 1's.

Links

• Andres Cicuttin, Several segments at different scales of the fractal walk starting at (0,0) with step of unit length turning right if a(n)=1 and left if a(n)=0, except when this rule determines a decrement in the horizontal axes which in that case the step is equal to previous one

Crossrefs

Conjectured quasiperiodicity in A271591 and A272170. A083035.

Programs

Mathematica

nmax=120 ; Table[If[Sin[2*Pi*(1/2)*n]+Sin[2*Pi*(1/(2*Sqrt[2]))*n]<0,0,1],{n,1,nmax}]

Formula

a(n) = floor( (1 + sin(2*Pi*(1/2)*n) + sin(2*Pi*(1/(2*Sqrt[2]))*n)) mod 2).