OFFSET
0,2
COMMENTS
We consider a one-dimensional automaton governed by the following rules:
- At stage 0, we have only one ON cell, at position z=0,
- An ON cell appears if it has exactly one ON neighbor:
+-------------+ +-----------+
| ...0(0)0... | |\ | ...(0)... |
| ...0(0)1... | --+ \ | ...(1)... |
| ...1(0)0... | --+ / | ...(1)... |
| ...1(0)1... | |/ | ...(0)... |
+-------------+ +-----------+
- An ON cell dies if its position and the number of its ON neighbors have a different parity:
+-----------+-----------+
| Even pos. | Odd pos. |
+-------------+ +-----------+-----------+
| ...0(1)0... | |\ | ...(1)... | ...(0)... |
| ...0(1)1... | --+ \ | ...(0)... | ...(1)... |
| ...1(1)0... | --+ / | ...(0)... | ...(1)... |
| ...1(1)1... | |/ | ...(1)... | ...(0)... |
+-------------+ +-----------+-----------+
Despite these simple rules, the evolution of the number of ON cells looks quite hectic.
The automaton depicted here is not a cellular automaton, as the evolution of a particular cell involves its position. However, by considering pairs of adjacent cells (say at position 2*z and 2*z+1), it is possible to represent this automaton by a 4-state cellular automaton.
Apparently, we obtain the Gould's sequence (A001316) by adding the following rule:
- An ON cell dies if it has no ON neighbor.
LINKS
Paul Tek, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Paul Tek, Illustration of the first 1000 stages
Paul Tek, PERL program for this sequence
EXAMPLE
After 0 generation:
- We have a unique ON cell at position z=0,
- Hence, a(0) = 1.
After 1 generation:
- ON cells appear at positions z=-1 and z=+1,
- No ON cell dies,
- Hence a(1) = a(0)+2-0 = 3.
After 2 generations:
- ON cells appears at positions z=-2 and z=+2,
- No ON cell dies,
- Hence a(2) = a(1)+2-0 = 5.
After 3 generations:
- ON cells appears at positions z=-3 and z=+3,
- ON cells at positions z=-1 and z=+1 die (as they have 2 ON neighbors),
- ON cells at positions z=-2 and z=+2 die (as they have 1 ON neighbor),
- Hence a(3) = a(2)+2-4 = 3.
Schematically:
+-----+-----------+------+
| n | ON cells | a(n) |
+-----+-----------+------+
| 0 | # | 1 |
| 1 | ### | 3 |
| 2 | ##### | 5 |
| 3 | # # # | 3 |
+=====+-----------+------+
| z%2 | 1010101 |
+-----+-----------+
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul Tek, Oct 11 2015
STATUS
approved