0,2

See A138276 for the corresponding sequence for a Bethe lattice with coordination number 3.

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.

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

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

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.

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

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 4. Rule 150 sums the values of the focal node and its k neighbors, then applies modulo 2.

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 and outmost nodes x_n are always 1.

Thus one has the patterns

x_0, x_1, x_2, ...

1

1 1

1 0 1

1 0 1 1

1 0 0 0 1

1 1 0 1 1 1

1 0 0 0 1 0 1

1 1 0 1 1 0 1 1

1 0 0 0 0 0 0 0 1

(N.B.: This is equivalent to the rght half plane of rule 150 in 1D.)

The nodes have the multiplicities 1,4,12,36,108,324,972,...

The sequence then is obtained by

a(n)= x_0(n) + 4*(x_1(n) + sum_(i=2...n) x_i(n) * 3^(i-1)

Cf. A138276, A072272, A007483, A071053, A001045.

Adjacent sequences: A138274 A138275 A138276 this_sequence A138278 A138279 A138280

Sequence in context: A120790 A025545 A082132 this_sequence A084601 A007231 A089794

nonn

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