A269696 - OEIS

%I #26 Jul 26 2024 21:16:31

%S 1,5,20,80,320,1280,5120,20480,81920,327680,1310720,5242880,20971520,

%T 83886080,335544320,1342177280

%N Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 6", based on the 5-celled von Neumann neighborhood.

%C Initialized with a single black (ON) cell at stage zero.

%C Rules 38, 70, 102, 134, 166, 198 and 230 also generate this sequence.

%C Apparently a duplicate of A003947. - _R. J. Mathar_, Mar 09 2016

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%F Conjectures from _Colin Barker_, Mar 08 2016: (Start)

%F a(n) = 5*4^(n-1) for n>0.

%F a(n) = 4*a(n-1) for n>1.

%F G.f.: (1+x) / (1-4*x).

%F (End)

%t rule=6; stages=300;

%t ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *)

%t on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)

%t Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)

%Y Cf. A269695.

%K nonn,easy

%O 0,2

%A _Robert Price_, Mar 03 2016

%E a(9)-a(15) from _Lars Blomberg_, Apr 12 2016