A289359 Image of 0 under repeated application of the morphism phi = {x -> x,x+1,x+2 if x mod 3 = 0; x -> x-1 if x mod 3 = 1; or x -> x+2 if x mod 3 = 2, for x = 0,1,2,3,...}.

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

Offset: 0

Author

Kerry Mitchell, Jul 03 2017

Keywords

• nonn

Comments

Similar to A288577, except the "x-1" term causes this sequence to return to 0 infinitely often, whereas A288577 is only 0 at the 0th term.

This is the limit of a series of iterations. The length of each iteration is given by A000213, beginning with the second term of that sequence (1, 3, 5, 8, 17, etc.):

0

0, 1, 2

0, 1, 2, 0, 4

0, 1, 2, 0, 4, 0, 1, 2, 3

0, 1, 2, 0, 4, 0, 1, 2, 3, 0, 1, 2, 0, 4, 3, 4, 5

When written in blocks in this way, each subsequent block is the concatenation of the previous block, the second-previous block, and the third-previous block, with each term incremented by 3.

Programs

Mathematica

SubstitutionSystem[{x_ -> Switch[Mod[x, 3], 0, {x, x+1, x+2}, 1, {x-1}, 2, {x+2}]}, {0}, 7] // Last (* Jean-François Alcover, Jan 21 2018 *)