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..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
Wolfram Research, Wolfram Atlas of Simple Programs
FORMULA
Conjectures from Chai Wah Wu, May 06 2024: (Start)
a(n) = 2*a(n-1) - 4*a(n-2) + 8*a(n-3) + a(n-8) - 2*a(n-9) + 4*a(n-10) - 8*a(n-11) for n > 32.
G.f.: (1048576*x^32 + 1310720*x^30 + 262144*x^28 + 327680*x^27 - 65536*x^26 - 49152*x^25 - 1118208*x^24 - 20480*x^23 - 1324032*x^22 - 5120*x^21 - 259072*x^20 - 327680*x^19 + 65280*x^18 + 48384*x^17 + 3840*x^16 + 19456*x^15 - 3328*x^14 - 256*x^13 - 256*x^12 - 976*x^11 - 64*x^10 - 252*x^9 + 64*x^8 + 49*x^7 + 5*x^6 + 5*x^5 + x^4 + x^3 + 5*x^2 + x + 1)/((x - 1)*(x + 1)*(2*x - 1)*(x^2 + 1)*(4*x^2 + 1)*(x^4 + 1)). (End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 782; 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[i, 2 * i - 1]], 2], {i , 1, stages - 1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 18 2017
STATUS
approved