A284177 - OEIS

%I #10 Mar 22 2017 12:19:20

%S 1,0,1,0,5,0,21,8,73,8,329,136,1161,168,5289,2216,18601,2728,84137,

%T 35496,297129,43688,1353897,567976,4761769,699048,21538985,9087656,

%U 76064937,11184808,346597545,145402536,1219012777,178956968,5513980073,2326440616

%N Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 809", based on the 5-celled von Neumann neighborhood.

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

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

%H Robert Price, <a href="/A284177/b284177.txt">Table of n, a(n) for n = 0..126</a>

%H Robert Price, <a href="/A284177/a284177.tmp.txt">Diagrams of first 20 stages</a>

%H 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

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

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

%H Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>

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

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

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

%F Conjectures from _Colin Barker_, Mar 22 2017: (Start)

%F G.f.: (1 + 4*x^4 + 16*x^6 + 8*x^7 - 204*x^8 + 256*x^10 + 128*x^11 - 192*x^12 + 32*x^13 + 32*x^14 + 512*x^17) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + 4*x^2)*(1 + 16*x^4)).

%F a(n) = a(n-2) + 256*a(n-8) - 256*a(n-10) for n>14.

%F (End)

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

%t code = 809; stages = 128;

%t rule = IntegerDigits[code, 2, 10];

%t g = 2 * stages + 1; (* Maximum size of grid *)

%t a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)

%t ca = a;

%t ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];

%t PrependTo[ca, a];

%t (* Trim full grid to reflect growth by one cell at each stage *)

%t k = (Length[ca[[1]]] + 1)/2;

%t ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];

%t Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 2], {i, 1, stages - 1}]

%Y Cf. A284175, A284176, A284178.

%K nonn,easy

%O 0,5

%A _Robert Price_, Mar 21 2017