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A265430 Number of OFF (white) cells in the n-th iteration of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.

%I A265430 #20 Feb 16 2025 08:33:27
%S A265430 0,1,3,3,5,6,8,8,10,11,13,13,15,16,18,18,20,21,23,23,25,26,28,28,30,
%T A265430 31,33,33,35,36,38,38,40,41,43,43,45,46,48,48,50,51,53,53,55,56,58,58,
%U A265430 60,61,63,63,65,66,68,68,70,71,73,73,75,76,78,78,80,81,83
%N A265430 Number of OFF (white) cells in the n-th iteration of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.
%D A265430 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A265430 Robert Price, <a href="/A265430/b265430.txt">Table of n, a(n) for n = 0..999</a>
%H A265430 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A265430 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A265430 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A265430 Conjectures from _Colin Barker_, Dec 09 2015 and Apr 16 2019: (Start)
%F A265430 a(n) = 1/8*(10*n+3*(-1)^n-(1-i)*(-i)^n-(1+i)*i^n-1) where i = sqrt(-1).
%F A265430 a(n) = a(n-1) + a(n-4) - a(n-5) for n>4.
%F A265430 G.f.: x*(1+2*x+2*x^3) / ((1-x)^2*(1+x)*(1+x^2)).
%F A265430 (End)
%e A265430 From _Michael De Vlieger_, Dec 09 2015: (Start)
%e A265430 First 12 rows, replacing "1" with "." for better visibility of OFF cells, with total number of 0's in the row to the right of the chart:
%e A265430                       .                        =  0
%e A265430                     . . 0                      =  1
%e A265430                   . 0 . 0 0                    =  3
%e A265430                 . . . . 0 0 0                  =  3
%e A265430               . . . 0 . 0 0 0 0                =  5
%e A265430             . . 0 . . . 0 0 0 0 0              =  6
%e A265430           . 0 . . . 0 . 0 0 0 0 0 0            =  8
%e A265430         . . . . 0 . . . 0 0 0 0 0 0 0          =  8
%e A265430       . . . 0 . . . 0 . 0 0 0 0 0 0 0 0        = 10
%e A265430     . . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0      = 11
%e A265430   . 0 . . . 0 . . . 0 . 0 0 0 0 0 0 0 0 0 0    = 13
%e A265430 . . . . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0 0 0  = 13
%e A265430 (End)
%t A265430 lim = 134; a = {}; Do[AppendTo[a, Take[#[[k]], 2 (k - 1) + 1]], {k, Floor[Length[#]/2]}] &@ CellularAutomaton[188, {{1}, 0}, lim]; Count[#, n_ /; n == 0] & /@ a (* _Michael De Vlieger_, Dec 09 2015 *)
%Y A265430 Cf. A118174.
%K A265430 nonn,easy
%O A265430 0,3
%A A265430 _Robert Price_, Dec 08 2015