A283407 - OEIS

%I #13 Feb 16 2025 08:33:42

%S 1,1,1,13,17,29,49,125,113,893,1137,2045,4081,8189,16369,32765,65521,

%T 131069,262129,524285,1048561,2097149,4194289,8388605,16777201,

%U 33554429,67108849,134217725,268435441,536870909,536870897,4294967293,8589934577,17179869181

%N Decimal 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 637", 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="/A283407/b283407.txt">Table of n, a(n) for n = 0..126</a>

%H Robert Price, <a href="/A283407/a283407.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="https://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) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n > 33.

%F G.f.: (-3221225472*x^33 + 536870912*x^32 + 3221225472*x^31 - 536870912*x^30 + 256*x^13 + 640*x^12 - 896*x^11 - 512*x^10 + 640*x^9 - 128*x^8 + 32*x^7 - 16*x^5 - 8*x^4 + 12*x^3 - 2*x^2 - x + 1)/((x - 1)*(x + 1)*(2*x - 1)). (End)

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

%t code = 637; 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]], 2], {i ,1, stages - 1}]

%Y Cf. A283404, A283405, A283406.

%K nonn,easy,changed

%O 0,4

%A _Robert Price_, Mar 07 2017