

A270459


Partial sums of the number of active (ON,black) cells in nth stage of growth of twodimensional cellular automaton defined by "Rule 165", based on the 5celled von Neumann neighborhood.


1



1, 9, 13, 53, 70, 174, 194, 379, 451, 763, 815, 1272, 1372, 1988, 2125, 2941, 3094, 4210, 4359, 5675, 5935, 7592, 7796, 9708, 10061, 12405, 12746, 15406, 15895, 18951, 19472, 22952, 23565, 27677, 28225, 32686, 33522, 38575, 39347, 44916, 45884, 52133, 53025
(list;
graph;
refs;
listen;
history;
text;
internal format)



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

Robert Price, Table of n, a(n) for n = 0..128
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5Neighbor Cellular Automata
Index to Elementary Cellular Automata


MATHEMATICA

CAStep[rule_, a_]:=Map[rule[[10#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=165; 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+1n, k1+n]], {j, k+1n, k1+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. A270457.
Sequence in context: A146039 A270234 A270718 * A270948 A270284 A270731
Adjacent sequences: A270456 A270457 A270458 * A270460 A270461 A270462


KEYWORD

nonn,easy


AUTHOR

Robert Price, Mar 17 2016


STATUS

approved



