A282985 - OEIS

%I #12 May 06 2024 15:45:59

%S 1,1,1,1001,1001,1001,1101001,10001001,1101001,1100001001,10111101001,

%T 11110001001,10001101001,10011100001001,10010111101001,10011110001001,

%U 11001010001101001,100101011100001001,10000010111101001,11011110011110001001,100010001010001101001

%N Binary 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 541", 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="/A282985/b282985.txt">Table of n, a(n) for n = 0..126</a>

%H Robert Price, <a href="/A282985/a282985.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 06 2024: (Start)

%F a(n) = 10001*a(n-4) - 10000*a(n-8) for n > 33.

%F G.f.: (-1000000000000000000000*x^33 - 910000000000000000000*x^32 + 899900091000000000000000000*x^31 + 8909991000000000000000000*x^30 - 10901099000000000000000000*x^29 + 990110000000000000000000*x^28 - 899889891000000000000000000*x^27 - 8920890900000000000000000*x^26 + 10901980100000000000000000*x^25 - 989089010000000000000000*x^24 - 10121010000000000000000*x^23 + 10901100000000000000000*x^22 - 990999000000000000000*x^21 - 10011000000000000000*x^20 + 10911100000000000000*x^19 - 90010000000000000*x^18 - 9000000000000*x^17 + 10901000000000000*x^16 - 101000000000000*x^15 - 91100000000000*x^14 - 990000000000*x^13 - 1000000000*x^12 - 88900000000*x^11 - 900000000*x^10 + 1090000000*x^9 - 8900000*x^8 - 10000*x^7 + 1091000*x^6 - 9000*x^5 - 9000*x^4 + 1001*x^3 + x^2 + x + 1)/(10000*x^8 - 10001*x^4 + 1). (End)

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

%t code = 541; 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[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

%Y Cf. A282984, A282986, A282987.

%K nonn,easy

%O 0,4

%A _Robert Price_, Feb 26 2017