A282261 - OEIS

%I #12 May 05 2024 17:05:20

%S 1,11,110,1111,11000,111000,1101010,11111111,110000010,1110000011,

%T 11010101010,111111111111,1100000010010,11100001010011,

%U 110101001110010,1111111001010011,11000001001110010,111000001001110011,1101010101001110010,11111111111001110011

%N Binary 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 446", 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="/A282261/b282261.txt">Table of n, a(n) for n = 0..126</a>

%H Robert Price, <a href="/A282261/a282261.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 _Chai Wah Wu_, May 05 2024: (Start)

%F a(n) = a(n-2) + 100000000*a(n-8) - 100000000*a(n-10) for n > 17.

%F G.f.: (100000*x^17 - 100000000*x^15 + 1100000*x^14 - 10101100*x^13 - 10091000*x^12 + 1111100*x^11 + 101000*x^10 - 1111100*x^9 + 8899000*x^8 + 11000111*x^7 + 1090010*x^6 + 109889*x^5 + 10890*x^4 + 1100*x^3 + 109*x^2 + 11*x + 1)/(100000000*x^10 - 100000000*x^8 - x^2 + 1). (End)

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

%t code = 446; 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]], 10], {i, 1, stages - 1}]

%Y Cf. A282262, A282263, A282264.

%K nonn,easy

%O 0,2

%A _Robert Price_, Feb 10 2017