A138277 - OEIS

%I #15 Aug 20 2018 05:20:56

%S 1,5,13,49,109,473,1081,4037,8749,37913,88465,325021,717337,3108461,

%T 7095613,26490289,57395629,248714393,580333585,2132141341,4707150193,

%U 20397650837,46548642709,173816036825,376630110937,1632063814061,3808148899477,13991111158153

%N Total number of active nodes of the Rule 150 cellular automaton on an infinite Bethe lattice with coordination number 4 (with a single 1 as initial condition).

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

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

%C Related to Cellular Automata.

%H Alois P. Heinz, <a href="/A138277/b138277.txt">Table of n, a(n) for n = 0..1000</a>

%H 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> [broken link]

%H 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 Jan Nagler and Jens Christian Claussen (2005), 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 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.

%e 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.

%e The focal node and outmost nodes x_n are always 1.

%e Thus one has the patterns

%e x_0, x_1, x_2, ...

%e 1

%e 1 1

%e 1 0 1

%e 1 0 1 1

%e 1 0 0 0 1

%e 1 1 0 1 1 1

%e 1 0 0 0 1 0 1

%e 1 1 0 1 1 0 1 1

%e 1 0 0 0 0 0 0 0 1

%e (N.B.: This is equivalent to the right half plane of Rule 150 in 1D.)

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

%e The sequence then is obtained by

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

%t nmax = 30;

%t states = CellularAutomaton[150, {{1}, 0}, nmax];

%t T[n_, i_] := states[[n+1, nmax+i+1]];

%t a[n_] := T[n, 0] + 4(T[n, 1]+Sum[3^(i-1) T[n, i], {i, 2, n}]);

%t Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Aug 20 2018 *)

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

%K nonn

%O 0,2

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

%E a(9)-a(27) from _Alois P. Heinz_, Jun 28 2015