A138276 Total number of active nodes of the Rule 150 cellular automaton on an infinite Bethe lattice with coordination number 3 (with a single 1 as initial condition).

1, 4, 6, 18, 30, 90, 102, 306, 510, 1530, 1542, 4626, 7110

Offset

0,2

Comments

See A138277 for the corresponding sequence for a Bethe lattice with coordination number 4.

See A001045 for the corresponding sequence on a 1D lattice (equivalent to a k=2 Bethe lattice); this is based on the Jacobsthal sequence A001045.

See A072272 for the corresponding sequence on a 2D lattice (based on A007483).

Related to Cellular Automata.

Links

Table of n, a(n) for n=0..12.

Jens Christian Claussen, Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration.

Jens Christian Claussen, Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration, arXiv:math.CO/0410429.

Jan Nagler and Jens Christian Claussen, 1/f^alpha spectra in elementary cellular automata and fractal signals, Phys. Rev. E 71 (2005), 067103

Formula

The total number of nodes in state 1 after n iterations (starting with a single 1) of the Rule 150 cellular automaton on an infinite Bethe lattice with coordination number 3. Rule 150 sums the values of the focal node and its k neighbors, then applies modulo 2.

Example

Let x_0 be the state (0 or 1) of the focal node and x_i the state of every node that is i steps away from the focal node. In time step n=0, all x_i=0 except x_0=1 (start with a single seed). In the next step, x_1=1 as they have 1 neighbor being 1. For n=2, the x_1 nodes have 1 neighbor being 1 (x_0) and
themselves being 1; the sum being 2, modulo 2, resulting in x_1=0. The focal node itself is 1 and has 3 neighbors being 1, sum being 4, modulo 2, resulting in x_0=0. The outmost nodes x_n are always 1.
Thus one has the patterns
x_0, x_1, x_2, ...
1
1 1
0 0 1
0 0 1 1
0 0 1 0 1
0 0 1 1 1 1
0 0 1 0 0 0 1
0 0 1 1 0 0 1 1
0 0 1 0 1 0 1 0 1
0 0 1 1 1 1 1 1 1 1
0 0 1 0 0 0 0 0 0 0 1
After 2 time steps, the x_0 and x_1 stay frozen at zero and the remaining x_i are generated by Rule 60 (or Rule 90 on half lattice spacing).
These nodes have multiplicities 1,3,6,12,24,48,96,192,384,768,...
The sequence then is obtained by
a(n)= x_0(n) + 3*(x_1(n) + sum_(i=2...n) x_i(n) * 2^(i-1)

Crossrefs

Cf. A138277, A072272, A007483, A071053, A001045.

Sequence in context: A064217 A026623 A026689 * A287682 A209236 A182643

Adjacent sequences: A138273 A138274 A138275 * A138277 A138278 A138279

Keyword

nonn

Author

Jens Christian Claussen (claussen(AT)theo-physik.uni-kiel.de), Mar 11 2008

Status

approved