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A273861 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 1006", based on the 5-celled von Neumann neighborhood. 1
1, 6, 15, 36, 65, 106, 167, 256, 357, 470, 603, 764, 953, 1186, 1495, 1872, 2269, 2678, 3107, 3564, 4049, 4578, 5183, 5856, 6565, 7318, 8139, 9044, 10049, 11218, 12631, 14200, 15805, 17422, 19059, 20724, 22417, 24154, 25967, 27848, 29765, 31726, 33755, 35868 (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

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

code=1006; 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. A169709.

Sequence in context: A273644 A273797 A273741 * A074132 A269543 A074149

Adjacent sequences:  A273858 A273859 A273860 * A273862 A273863 A273864

KEYWORD

nonn,easy

AUTHOR

Robert Price, Jun 01 2016

STATUS

approved

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Last modified October 22 18:45 EDT 2021. Contains 348175 sequences. (Running on oeis4.)