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A273744 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 901", based on the 5-celled von Neumann neighborhood. 1
1, 9, 34, 83, 164, 285, 454, 679, 968, 1329, 1770, 2299, 2924, 3653, 4494, 5455, 6544, 7769, 9138, 10659, 12340, 14189, 16214, 18423, 20824, 23425, 26234, 29259, 32508, 35989, 39710, 43679, 47904, 52393, 57154, 62195, 67524, 73149, 79078, 85319, 91880, 98769 (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
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
FORMULA
Conjectures from Colin Barker, May 29 2016: (Start)
a(n) = n*(11+12*n+4*n^2)/3 for n>0.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
G.f.: (1+5*x+4*x^2-3*x^3+x^4) / (1-x)^4.
(End)
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=901; 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. A273743.
Sequence in context: A262959 A139275 A236370 * A133547 A100179 A106598
KEYWORD
nonn,easy
AUTHOR
Robert Price, May 28 2016
STATUS
approved

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Last modified May 28 01:34 EDT 2024. Contains 372900 sequences. (Running on oeis4.)