NAME
Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 446", based on the 5-celled von Neumann neighborhood.
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Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 446", based on the 5-celled von Neumann neighborhood.
(history; published version)
(history; published version)
STATUS
reviewed
approved
STATUS
proposed
reviewed
STATUS
editing
proposed
LINKS
Robert Price, <a href="/A272250/b272250.txt">Table of n, a(n) for n = 0..128</a>
STATUS
approved
editing
STATUS
proposed
approved
STATUS
editing
proposed
NAME
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 446", based on the 5-celled von Neumann neighborhood.
DATA
1, 6, 18, 43, 71, 123, 187, 296, 372, 513, 665, 898, 1090, 1371, 1683, 2076, 2360, 2805, 3261, 3886, 4390, 5079, 5799, 6668, 7352, 8333, 9245, 10414, 11498, 12711, 14091, 15560, 16820, 18453, 20089, 22062, 23786, 25847, 27963, 30372, 32380, 34965, 37385
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=446; 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 *)
CROSSREFS
Cf. A272194.
KEYWORD
allocated
nonn,easy
AUTHOR
Robert Price, Apr 23 2016
STATUS
approved
editing
NAME
allocated for Robert Price
KEYWORD
allocated
STATUS
approved