Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 84", based on the 5-celled von Neumann neighborhood.

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Robert Price, <a href="/A270106/b270106.txt">Table of n, a(n) for n = 0..128</a>

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allocated for Robert PricePartial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 84", based on the 5-celled von Neumann neighborhood.

1, 5, 13, 29, 45, 77, 109, 173, 205, 269, 333, 461, 525, 653, 781, 1037, 1101, 1229, 1357, 1613, 1741, 1997, 2253, 2765, 2893, 3149, 3405, 3917, 4173, 4685, 5197, 6221, 6349, 6605, 6861, 7373, 7629, 8141, 8653, 9677, 9933, 10445, 10957, 11981, 12493, 13517

0,2

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

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

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

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

<a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

<a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>

<a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];

code=84; stages=128;

rule=IntegerDigits[code, 2, 10];

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

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

ca=a;

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

PrependTo[ca, a];

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

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

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

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

Table[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum at each stage *)

Cf. A189007.

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Robert Price, Mar 11 2016

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allocated for Robert Price

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