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A270335 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 158", based on the 5-celled von Neumann neighborhood. 1
1, 6, 18, 38, 70, 102, 166, 218, 322, 382, 534, 642, 842, 1018, 1286, 1522, 1850, 2110, 2562, 2854, 3374, 3762, 4418, 4898, 5682, 6198, 7106, 7754, 8850, 9574, 10778, 11694, 12934, 13934, 15386, 16474, 18006, 19322, 21114, 22482, 24458, 26094, 28302, 30034 (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 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

MATHEMATICA

(* From Robert Price, Start *)

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

code = 158; 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 *)

(* From Robert Price, End *)

Accumulate[Total[#, 2] & /@ CellularAutomaton[{158, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, {20}]] (* JungHwan Min, Mar 16 2016 *)

CROSSREFS

Cf. A270333.

Sequence in context: A005899 A261652 A180118 * A270940 A270081 A261651

Adjacent sequences:  A270332 A270333 A270334 * A270336 A270337 A270338

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 15 2016

STATUS

approved

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Last modified February 24 12:52 EST 2018. Contains 299623 sequences. (Running on oeis4.)