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
Keywords
Examples
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 * Sum_{i=1..n} x_i(n) * 2^(i-1)
Links
- 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.
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