OFFSET
0,4
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..126
Robert Price, Diagrams of 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
FORMULA
Conjectures from Colin Barker, Nov 30 2016: (Start)
a(n) = 101*a(n-2) - 10100*a(n-6) + 10000*a(n-8) for n>15.
G.f.: (1 +x -100*x^2 +1010*x^3 -x^4 -2100*x^5 -1000*x^7 +10200*x^8 +100000*x^9 -10000*x^10 +100000*x^11 -1010000*x^12 -10100000*x^13 +1000000*x^14 +10000000*x^15) / ((1 -x)*(1 +x)*(1 -10*x)*(1 +10*x)*(1 -10*x^2)*(1 +10*x^2)).
(End)
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=107; 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}];
Table[FromDigits[Part[ca[[i]][[i]], Range[1, i]], 10], {i, 1, stages-1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Nov 30 2016
STATUS
approved