A278899 - OEIS

%I #9 Nov 30 2016 13:00:44

%S 1,10,100,1111,100,111011,0,11101111,1010000,1110101111,101000000,

%T 111011111111,10101000000,11101010111111,1010100000000,

%U 1110101111111111,101010100000000,111010101011111111,10101010000000000,11101010111111111111,1010101010000000000

%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 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="/A278899/b278899.txt">Table of n, a(n) for n = 0..126</a>

%H Robert Price, <a href="/A278899/a278899.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 <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_, Nov 30 2016: (Start)

%F a(n) = 101*a(n-2) - 10100*a(n-6) + 10000*a(n-8) for n>15.

%F G.f.: (1 +10*x -x^2 +101*x^3 -10000*x^4 -1200*x^5 -10000*x^7 +2010000*x^8 +10000*x^9 -1000000*x^10 +1000000*x^11 -101000000*x^12 -101000000*x^13 +100000000*x^14 +100000000*x^15) / ((1 -x)*(1 +x)*(1 -10*x)*(1 +10*x)*(1 -10*x^2)*(1 +10*x^2)).

%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. A278898, A278900, A278901.

%K nonn,easy

%O 0,2

%A _Robert Price_, Nov 30 2016