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A290073 Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 641", based on the 5-celled von Neumann neighborhood. 4
1, 2, 5, 12, 29, 60, 125, 252, 509, 1020, 2045, 4092, 8189, 16380, 32765, 65532, 131069, 262140, 524285, 1048572, 2097149, 4194300, 8388605, 16777212, 33554429, 67108860, 134217725, 268435452, 536870909, 1073741820, 2147483645, 4294967292, 8589934589 (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, Jul 20 2017: (Start)
G.f.: (1 + 2*x^3 + 4*x^4) / ((1 - x)*(1 + x)*(1 - 2*x)).
a(n) = 2^(n+1) - 3 = A283506(n) for n>1 and even.
a(n) = 2^(n+1) - 4 = A283506(n) for n>1 and odd.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>4.
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 641; 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]], 10], {i, 1, stages - 1}]
CROSSREFS
Sequence in context: A326761 A162036 A321253 * A232534 A274594 A062422
KEYWORD
nonn,easy
AUTHOR
Robert Price, Jul 19 2017
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)