A335933 A fractal function, related to ruler functions. a(1) = 0; otherwise for m >= 0, a(3m) = 1, a(3m-1) = a(2m-1) + sign(a(2m-1)), a(3m+1) = a(2m+1) + sign(a(2m+1)).

1, 0, 0, 1, 2, 2, 1, 3, 3, 1, 4, 4, 1, 2, 2, 1, 5, 5, 1, 3, 3, 1, 2, 2, 1, 6, 6, 1, 4, 4, 1, 2, 2, 1, 3, 3, 1, 7, 7, 1, 2, 2, 1, 5, 5, 1, 3, 3, 1, 2, 2, 1, 4, 4, 1, 8, 8, 1, 2, 2, 1, 3, 3, 1, 6, 6, 1, 2, 2, 1, 4, 4, 1, 3, 3, 1, 2, 2, 1, 5, 5, 1

Offset: 0

Author

Peter Munn, Jun 30 2020

Keywords

• nonn
• easy

Comments

We choose a form for the definition that shows clearly its relationship to A307744.

The odd bisection is essentially A087088.

If we add a(-1) = 0 to the definition and allow negative m (and therefore n), we get a symmetric function, that is a(n) = a(-n).

For k >= 1 numbers 1..k occur with the same periodic and mirror symmetries as in A307744 and in ruler function A051064. In A051064, k occurs 3 times more frequently than k+1. Here, and in A307744, k occurs 3/2 times more frequently than k+1, precisely 2^(k-1) times in every 3^k terms.

Crossrefs

Sequences with similar definitions: A205593, A307744.

A051064 has matching symmetries.

Odd bisection: A087088.

Programs

PARI

a(n) = if (n==1, 0, if ((n%3) == 0, 1, if ((n%3)==1, my(k=(n-1)/3, aa = a(2*k+1)); aa+sign(aa),  my(k=(n+1)/3, aa = a(2*k-1)); aa+sign(aa)))); \\ Michel Marcus, Jul 03 2020