A151895 Number of ON cells after n generations of the cellular automaton on the square grid that is described in the Comments.

0, 1, 5, 9, 13, 25, 29, 41, 53, 65, 85, 97, 117, 145, 149, 161, 173, 185, 213, 233, 261, 297, 333, 385, 429, 481, 533, 545, 573, 601, 629, 673, 717, 761, 837, 905, 989, 1033, 1085, 1145, 1197, 1257, 1309, 1337, 1397, 1457, 1525, 1625, 1669

Offset: 0

Author

David Applegate and N. J. A. Sloane, Jul 30 2009

Keywords

• nonn

Comments

The cells are the squares of the standard square grid.

Cells are either OFF or ON, once they are ON they stay ON, and we begin in generation 1 with 1 ON cell.

Each cell has 4 neighbors, those that it shares an edge with. Cells that are ON at generation n all try simultaneously to turn ON all their neighbors that are OFF. They can only do this at this point in time; afterwards they go to sleep (but stay ON).

A square Q is turned ON at generation n+1 if:

a) Q shares an edge with one and only one square P (say) that was turned ON at generation n (in which case the two squares which intersect Q only in a vertex not on that edge are called Q's "outer squares"), and

b) Q's outer squares were not considered (that is, satisfied a)) in any previous generation, and

c) Q's outer squares are not prospective squares of the (n+1)st generation satisfying a).

Originally constructed in an attempt to explain the Holladay-Ulam CA shown in Fig. 2 of the 1962 Ulam article. However, as explained on page 222 of that article, the actual rule for that CA (see A151906, A151907) is different from ours.

A170896 and A267190 are also closely related cellular automata.

A151895 and A267190 first differ at n=17, when A267190 turns (12,2) ON even though its outer square (11,1) was considered (not turned ON) in a previous generation. - David Applegate, Jan 30 2016

References

• D. Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191.

Links

• David Applegate, Table of n, a(n) for n = 0..250
• David Applegate, The movie version
• David Applegate, Illustration of first 10 generations
• David Applegate, Illustration of first 20 generations
• David Applegate, Illustration of first 32 generations
• David Applegate, Illustration of first 64 generations
• David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.], which is also available at arXiv:1004.3036v2
• R. G. Schrandt and S. M. Ulam, On recursively defined geometric objects and patterns of growth [Link supplied by Laurinda J. Alcorn, Jan 09 2010.]
• N. J. A. Sloane, Illustration of initial terms (concentrating on a 90-degree sector)
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 [Annotated scanned copy]

Crossrefs

See A170896, A170897 for the original Schrandt-Ulam version.

Cf. A151896 (the first differences), A139250, A151905, A151906, A151907, A267190, A267191.

Formula

We do not know of a recurrence or generating function.

Extensions

Entry (including definition) revised by David Applegate and N. J. A. Sloane, Jan 21 2016