A285835 - OEIS

%I #11 Feb 16 2025 08:33:44

%S 1,3,6,15,12,63,24,255,240,1023,992,4095,4032,16383,16256,65535,65280,

%T 262143,261632,1048575,1047552,4194303,4192256,16777215,16773120,

%U 67108863,67100672,268435455,268419072,1073741823,1073709056,4294967295,4294901760

%N Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 107", 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="/A285835/b285835.txt">Table of n, a(n) for n = 0..126</a>

%H Robert Price, <a href="/A285835/a285835.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 _Colin Barker_, Apr 28 2017: (Start)

%F G.f.: (1+3*x-x^2-6*x^3-16*x^4+16*x^6+192*x^8-448*x^10+256*x^12) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-2*x^2)).

%F a(n) = 2^n - 2^(n/2) for n>6 and even.

%F a(n) = 2^(n + 1) - 1 for n>6 and odd.

%F a(n) = 7*a(n-2) - 14*a(n-4) + 8*a(n-6) for n>6.

%F (End)

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

%t code = 107; 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. A284940, A285833, A285834.

%K nonn,easy,changed

%O 0,2

%A _Robert Price_, Apr 27 2017