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A270894

First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 209", based on the 5-celled von Neumann neighborhood.
(history; published version)

#8 by Charles R Greathouse IV at Fri Jul 26 21:16:35 EDT 2024

NAME

First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 209", 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 Sat Mar 26 11:33:56 EDT 2016

STATUS

proposed

approved

#6 by Robert Price at Sat Mar 26 10:53:19 EDT 2016

STATUS

editing

proposed

#5 by Robert Price at Sat Mar 26 10:53:16 EDT 2016

LINKS

Robert Price, <a href="/A270894/b270894.txt">Table of n, a(n) for n = 0..127</a>

STATUS

approved

editing

#4 by N. J. A. Sloane at Fri Mar 25 23:02:46 EDT 2016

STATUS

proposed

approved

#3 by Robert Price at Fri Mar 25 11:08:57 EDT 2016

STATUS

editing

proposed

#2 by Robert Price at Fri Mar 25 11:08:51 EDT 2016

NAME

allocated for Robert PriceFirst differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 209", based on the 5-celled von Neumann neighborhood.

DATA

3, 5, 19, -3, 55, -27, 83, -43, 139, -91, 211, -163, 299, -219, 331, -243, 435, -363, 659, -667, 1003, -983, 1163, -947, 1083, -891, 1227, -1143, 1495, -1383, 1635, -1491, 1887, -1755, 2371, -2475, 3155, -3099, 3411, -3203, 3663, -3495, 4059, -3951, 4503

OFFSET

0,1

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=209; 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[on[[i+1]]-on[[i]], {i, 1, Length[on]-1}] (* Difference at each stage *)

CROSSREFS

Cf. A270891.

KEYWORD

allocated

sign,easy

AUTHOR

Robert Price, Mar 25 2016

STATUS

approved

editing

#1 by Robert Price at Fri Mar 25 11:08:51 EDT 2016

NAME

allocated for Robert Price

KEYWORD

allocated

STATUS

approved