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a(n) = (-2-(-2)^n+2*(-1)^n+5*2^n)/4.

a(n) = (-2-(-2)^n+2*(-1)^n+5*2^n)/4. (End)

proposed

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proposed

Conjectures from Colin Barker, Nov 03 2016: (Start)

a(n) = (-2-(-2)^n+2*(-1)^n+5*2^n)/4.

G.f.: (1+2*x-x^2+x^3) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)).

(End)

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proposed

Cf. A277864, A277865, A277866.

Robert Price, <a href="/A277867/a277867.tmp.txt">Diagrams of first 20 stages</a>

Robert Price, <a href="/A277867/a277867.tmp.txt">Diagrams of first 20 stages</a>

Robert Price, <a href="/A277867/b277867.txt">Table of n, a(n) for n = 0..126</a>

Robert Price, <a href="/A277867/a277867.tmp.txt">Diagrams of first 20 stages</a>

allocated for Robert PriceDecimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 3", based on the 5-celled von Neumann neighborhood.

1, 2, 4, 11, 16, 47, 64, 191, 256, 767, 1024, 3071, 4096, 12287, 16384, 49151, 65536, 196607, 262144, 786431, 1048576, 3145727, 4194304, 12582911, 16777216, 50331647, 67108864, 201326591, 268435456, 805306367, 1073741824, 3221225471, 4294967296, 12884901887

0,2

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

<a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

<a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>

<a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];

code=3; 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}]

allocated

nonn,easy

Robert Price, Nov 02 2016

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