A269872 - OEIS

%I #10 Jul 26 2024 21:16:31

%S 1,4,5,32,13,84,29,168,33,268,61,416,53,580,121,748,105,964,185,1188,

%T 181,1500,221,1784,221,2108,305,2480,317,2812,437,3188,421,3616,545,

%U 4152,469,4576,645,5124,521,5712,813,6100,801,6784,885,7356,917,8076,1005

%N Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 41", based on the 5-celled von Neumann neighborhood.

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

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

%H Robert Price, <a href="/A269872/b269872.txt">Table of n, a(n) for n = 0..100</a>

%H Robert Price, <a href="/A269872/a269872.tmp.txt">Diagrams of the first 20 stages.</a>

%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>

%t code=41; stages=100;

%t rule=IntegerDigits[code,2,10];

%t g=2*stages+1; (* Maximum size of grid *)

%t a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)

%t ca=a;

%t ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];

%t PrependTo[ca,a];

%t (* Trim full grid to reflect growth by one cell at each stage *)

%t k=(Length[ca[[1]]]+1)/2;

%t ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];

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

%K nonn,easy

%O 0,2

%A _Robert Price_, Mar 06 2016