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%I #7 Jan 21 2018 09:36:29
%S 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,
%T 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,
%U 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
%N 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,...}.
%C 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.
%C 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.):
%C 0
%C 0, 1, 2
%C 0, 1, 2, 0, 4
%C 0, 1, 2, 0, 4, 0, 1, 2, 3
%C 0, 1, 2, 0, 4, 0, 1, 2, 3, 0, 1, 2, 0, 4, 3, 4, 5
%C 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.
%t 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 *)
%K nonn
%O 0,3
%A _Kerry Mitchell_, Jul 03 2017
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