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A270678

Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 189", based on the 5-celled von Neumann neighborhood.
(history; published version)

#8 by Charles R Greathouse IV at Fri Jul 26 21:16:34 EDT 2024

NAME

Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 189", based on the 5-celled von Neumann neighborhood.

Discussion

Fri Jul 26

21:16

OEIS Server: https://oeis.org/edit/global/2997

#7 by Joerg Arndt at Fri Jun 03 06:02:04 EDT 2016

STATUS

proposed

approved

#6 by Lars Blomberg at Fri Jun 03 05:27:44 EDT 2016

STATUS

editing

proposed

#5 by Lars Blomberg at Fri Jun 03 05:21:45 EDT 2016

DATA

1, 8, 44, 212, 940, 3857, 15648, 62928, 251952, 1005696, 4021440, 16072420, 64261572, 256960036, 1027642244, 4110139460

EXTENSIONS

a(8)-a(15) from Lars Blomberg, Jun 03 2016

STATUS

approved

editing

#4 by N. J. A. Sloane at Mon Mar 21 21:15:20 EDT 2016

STATUS

proposed

approved

#3 by Robert Price at Mon Mar 21 11:16:53 EDT 2016

STATUS

editing

proposed

#2 by Robert Price at Mon Mar 21 11:16:49 EDT 2016

NAME

allocated for Robert PriceNumber of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 189", based on the 5-celled von Neumann neighborhood.

DATA

1, 8, 44, 212, 940, 3857, 15648, 62928

OFFSET

0,2

COMMENTS

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

REFERENCES

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

LINKS

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>

MATHEMATICA

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

code=189; 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 *)

Part[on, 2^Range[0, Log[2, stages]]] (* Extract relevant terms *)

CROSSREFS

Cf. A270677.

KEYWORD

allocated

nonn,more

AUTHOR

Robert Price, Mar 21 2016

STATUS

approved

editing

#1 by Robert Price at Mon Mar 21 11:16:49 EDT 2016

NAME

allocated for Robert Price

KEYWORD

allocated

STATUS

approved