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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).

%I A138276 #10 Jan 31 2025 03:49:46
%S A138276 1,4,6,18,30,90,102,306,510,1530,1542,4626,7110
%N 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).
%C A138276 See A138277 for the corresponding sequence for a Bethe lattice with coordination number 4.
%C A138276 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.
%C A138276 See A072272 for the corresponding sequence on a 2D lattice (based on A007483).
%C A138276 Related to Cellular Automata.
%H A138276 Jens Christian Claussen, <a href="http://www.theo-physik.uni-kiel.de/%7Eclaussen/rule150.pdf">Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration</a>.
%H A138276 Jens Christian Claussen, <a href="http://arXiv.org/abs/math.CO/0410429">Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration</a>, arXiv:math.CO/0410429.
%H A138276 Jan Nagler and Jens Christian Claussen, 1/f^alpha spectra in elementary cellular automata and fractal signals, <a href="http://dx.doi.org/10.1103/PhysRevE.71.067103">Phys. Rev. E 71 (2005), 067103</a>
%F A138276 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.
%e A138276 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
%e A138276 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.
%e A138276 Thus one has the patterns
%e A138276 x_0, x_1, x_2, ...
%e A138276 1
%e A138276 1 1
%e A138276 0 0 1
%e A138276 0 0 1 1
%e A138276 0 0 1 0 1
%e A138276 0 0 1 1 1 1
%e A138276 0 0 1 0 0 0 1
%e A138276 0 0 1 1 0 0 1 1
%e A138276 0 0 1 0 1 0 1 0 1
%e A138276 0 0 1 1 1 1 1 1 1 1
%e A138276 0 0 1 0 0 0 0 0 0 0 1
%e A138276 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).
%e A138276 These nodes have multiplicities 1,3,6,12,24,48,96,192,384,768,...
%e A138276 The sequence then is obtained by
%e A138276 a(n) = x_0(n) + 3 * Sum_{i=1..n} x_i(n) * 2^(i-1)
%Y A138276 Cf. A138277, A072272, A007483, A071053, A001045.
%K A138276 nonn
%O A138276 0,2
%A A138276 Jens Christian Claussen (claussen(AT)theo-physik.uni-kiel.de), Mar 11 2008