cp's OEIS Frontend

A269716 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 22", based on the 5-celled von Neumann neighborhood.

%I A269716 #18 Feb 16 2025 08:33:30
%S A269716 1,5,20,88,368,1504,6080,24448,98048,392704,1571840,6289408,25161728,
%T A269716 100655104,402636800,1610579968
%N A269716 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 22", based on the 5-celled von Neumann neighborhood.
%C A269716 Initialized with a single black (ON) cell at stage zero.
%C A269716 Appears to coincide with A093357 after the second term. - _R. J. Mathar_, Mar 09 2016
%D A269716 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A269716 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 A269716 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A269716 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A269716 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A269716 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A269716 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A269716 Conjectures from _Colin Barker_, Mar 08 2016: (Start)
%F A269716 a(n) = 2^(n-1)*(3*2^n-2) for n>1.
%F A269716 a(n) = 6*a(n-1)-8*a(n-2) for n>3.
%F A269716 G.f.: (1+2*x)*(1-3*x+4*x^2) / ((1-2*x)*(1-4*x)).
%F A269716 (End)
%t A269716 rule=22; stages=300;
%t A269716 ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *)
%t A269716 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A269716 Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)
%Y A269716 Cf. A269715.
%K A269716 nonn,more
%O A269716 0,2
%A A269716 _Robert Price_, Mar 04 2016
%E A269716 a(9)-a(15) from _Lars Blomberg_, Apr 15 2016