A288577 -id:A288577 - cp's OEIS frontend

A289281 Square array whose rows m >= 2 hold the limit under iterations of the morphism { x -> (x, ..., x+k-1) if k|x ; x -> x+1 otherwise }, starting with (0); read by falling antidiagonals.

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

Offset: 2

M. F. Hasler, Jul 01 2017

The generalization of A104234 (row 2) and A288577 (row 3) to arbitrary m.

			The array starts (first row: m=2)
  [ 0 1 2 2 3 2 3 4 2 3  4  4  5  2  3  4  4  5  4  5  6  2  3  4  4 ...]
  [ 0 1 2 2 3 3 3 4 5 3  4  5  3  4  5  5  6  3  4  5  5  6  3  4  5 ...]
  [ 0 1 2 3 2 3 4 3 4 4  5  6  7  4  4  5  6  7  4  5  6  7  6  7  8 ...]
  [ 0 1 2 3 4 2 3 4 5 3  4  5  5  6  7  8  9  4  5  5  6  7  8  9  5 ...]
  [ 0 1 2 3 4 5 2 3 4 5  6  3  4  5  6  6  7  8  9 10 11  4  5  6  6 ...]
  [ 0 1 2 3 4 5 6 2 3 4  5  6  7  3  4  5  6  7  7  8  9 10 11 12 13 ...]
  [ 0 1 2 3 4 5 6 7 2 3  4  5  6  7  8  3  4  5  6  7  8  8  9 10 11 ...]
  [ 0 1 2 3 4 5 6 7 8 2  3  4  5  6  7  8  9  3  4  5  6  7  8  9  9 ...]
  [ 0 1 2 3 4 5 6 7 8 9  2  3  4  5  6  7  8  9 10  3  4  5  6  7  8 ...]
  [ 0 1 2 3 4 5 6 7 8 9 10  2  3  4  5  6  7  8  9 10 11  3  4  5  6 ...]
  [ 0 1 2 3 4 5 6 7 8 9 10 11  2  3  4  5  6  7  8  9 10 11 12  3  4 ...]
  [ 0 1 2 3 4 5 6 7 8 9 10 11 12  2  3  4  5  6  7  8  9 10 11 12 13 ...]
  ...
It is easy to prove that row m starts with (0, ..., m-1; 2, ..., m; 3, ..., m; m, ..., 2m-1; ...).

Kerry Mitchell, Table of n, a(n) for n = 2..10012

Cf. A104234 (row 2), A288577 (row 3).

A289281_row(n=30,k=2,a=[0])={while(#a

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,...}.

Original entry on oeis.org

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

Kerry Mitchell, Jul 03 2017

nonn

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.

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 *)

cp's OEIS Frontend

A289281 Square array whose rows m >= 2 hold the limit under iterations of the morphism { x -> (x, ..., x+k-1) if k|x ; x -> x+1 otherwise }, starting with (0); read by falling antidiagonals.

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