A273329 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 646", based on the 5-celled von Neumann neighborhood.

4, 0, 12, -8, 20, -12, 44, -44, 60, -44, 92, -80, 116, -92, 188, -196, 220, -196, 268, -256, 292, -268, 364, -364, 412, -364, 508, -472, 580, -508, 796, -828, 876, -828, 972, -960, 996, -972, 1068, -1068, 1116, -1068, 1212, -1176, 1284, -1212, 1500, -1524

Offset: 0

Author

Robert Price, May 20 2016

Keywords

• sign
• easy

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..127
• 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

Crossrefs

Cf. A273326.

Programs

Mathematica

CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=646; 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[on[[i+1]]-on[[i]],{i,1,Length[on]-1}] (* Difference at each stage *)