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%I #34 Sep 02 2019 02:30:12
%S 1,3,5,9,13,18,23,30,37,45,53,63,73,84,95,108,121,135,149,165,181,198,
%T 215,234,253,273,293,315,337,360,383,408,433,459,485,513,541,570,599,
%U 630,661,693,725,759,793,828,863,900,937,975,1013,1053,1093,1134
%N Total number of ON (black) cells after n iterations of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.
%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H Robert Price, <a href="/A265429/b265429.txt">Table of n, a(n) for n = 0..999</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</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_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F Conjectures from _Colin Barker_, Dec 09 2015 and Apr 16 2019: (Start)
%F a(n) = (1/16)*(6*n^2 + 24*n - 3*(-1)^n + 2*(-i)^n + 2*i^n + 15) where i = sqrt(-1).
%F G.f.: (1 + x + 2*x^3 - x^4) / ((1-x)^3*(1+x)*(1+x^2)).
%F (End)
%F Conjecture: the sequence consists of all numbers k > 0 such that floor(sqrt(8*(k+1)/3)) != floor(sqrt(8*k/3)). - _Gevorg Hmayakyan_, Sep 01 2019
%e From _Michael De Vlieger_, Dec 09 2015: (Start)
%e First 12 rows, replacing "0" with ".", ignoring "0" outside of range of 1's for better visibility of ON cells, followed by total number of ON cells per row, and running total up to that row:
%e 1 = 1 -> 1
%e 1 1 = 2 -> 3
%e 1 . 1 = 2 -> 5
%e 1 1 1 1 = 4 -> 9
%e 1 1 1 . 1 = 4 -> 13
%e 1 1 . 1 1 1 = 5 -> 18
%e 1 . 1 1 1 . 1 = 5 -> 23
%e 1 1 1 1 . 1 1 1 = 7 -> 30
%e 1 1 1 . 1 1 1 . 1 = 7 -> 37
%e 1 1 . 1 1 1 . 1 1 1 = 8 -> 45
%e 1 . 1 1 1 . 1 1 1 . 1 = 8 -> 53
%e 1 1 1 1 . 1 1 1 . 1 1 1 = 10 -> 63
%e 1 1 1 . 1 1 1 . 1 1 1 . 1 = 10 -> 72
%e (End)
%t rule = 188; rows = 30; Table[Total[Take[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}],k]],{k,1,rows}]
%t Accumulate[Count[#, n_ /; n == 1] & /@ CellularAutomaton[188, {{1}, 0}, 53]] (* _Michael De Vlieger_, Dec 09 2015 *)
%Y Cf. A118174.
%K nonn,easy
%O 0,2
%A _Robert Price_, Dec 08 2015
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