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A265428 Number of ON (black) cells in the n-th iteration of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.

%I A265428 #27 Feb 16 2025 08:33:27
%S A265428 1,2,2,4,4,5,5,7,7,8,8,10,10,11,11,13,13,14,14,16,16,17,17,19,19,20,
%T A265428 20,22,22,23,23,25,25,26,26,28,28,29,29,31,31,32,32,34,34,35,35,37,37,
%U A265428 38,38,40,40,41,41,43,43,44,44,46,46,47,47,49,49,50,50
%N A265428 Number of ON (black) cells in the n-th iteration of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.
%D A265428 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A265428 Robert Price, <a href="/A265428/b265428.txt">Table of n, a(n) for n = 0..999</a>
%H A265428 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A265428 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A265428 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A265428 Conjectures from _Colin Barker_, Dec 09 2015 and Apr 16 2019: (Start)
%F A265428 a(n) = (1/8)*(6*n-3*(-1)^n+(1-i)*(-i)^n+(1+i)*i^n+9) where i = sqrt(-1).
%F A265428 a(n) = a(n-1) + a(n-4) - a(n-5) for n > 4.
%F A265428 G.f.: (1+x+2*x^3-x^4) / ((1-x)^2*(1+x)*(1+x^2)). (End)
%F A265428 a(n) = (6*n+9+2*cos(n*Pi/2)-3*cos(n*Pi)-2*sin(n*Pi/2))/8. - _Wesley Ivan Hurt_, Oct 02 2017
%e A265428 From _Michael De Vlieger_, Dec 09 2015: (Start)
%e A265428 First 12 rows, replacing "0" with ".", ignoring "0" outside of range of 1's for better visibility of ON cells, with total number of 1's in the row to the left of the chart:
%e A265428 1  =   1
%e A265428 2  =   1 1
%e A265428 2  =   1 . 1
%e A265428 4  =   1 1 1 1
%e A265428 4  =   1 1 1 . 1
%e A265428 5  =   1 1 . 1 1 1
%e A265428 5  =   1 . 1 1 1 . 1
%e A265428 7  =   1 1 1 1 . 1 1 1
%e A265428 7  =   1 1 1 . 1 1 1 . 1
%e A265428 8  =   1 1 . 1 1 1 . 1 1 1
%e A265428 8  =   1 . 1 1 1 . 1 1 1 . 1
%e A265428 10 =   1 1 1 1 . 1 1 1 . 1 1 1
%e A265428 10 =   1 1 1 . 1 1 1 . 1 1 1 . 1
%e A265428 (End)
%t A265428 rule = 188; rows = 30; Table[Total[Table[Take[CellularAutomaton[rule, {{1},0},rows-1,{All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}]
%t A265428 Count[#, n_ /; n == 1] & /@ CellularAutomaton[188, {{1}, 0}, 66] (* _Michael De Vlieger_, Dec 09 2015 *)
%Y A265428 Cf. A118174.
%K A265428 nonn,easy
%O A265428 0,2
%A A265428 _Robert Price_, Dec 08 2015