A273430 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 825", based on the 5-celled von Neumann neighborhood.

1, 4, 17, 28, 61, 93, 141, 157, 221, 296, 393, 405, 480, 577, 693, 733, 857, 1025, 1145, 1297, 1397, 1597, 1637, 1765, 1965, 2145, 2229, 2433, 2689, 2681, 2981, 3081, 3389, 3537, 3985, 4381, 4413, 4781, 4969, 5105, 5421, 5693, 6073, 6165, 6525, 6725, 7101

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

Robert Price, Diagrams of the first 20 stages

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=825; 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}];
Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)

Crossrefs

Sequence in context: A272769 A273646 A272991 * A273073 A272330 A163736

Adjacent sequences: A273427 A273428 A273429 * A273431 A273432 A273433

Keyword

nonn,easy

Author

Robert Price, May 27 2016

Status

approved