A277952 - OEIS

%I #16 Nov 05 2025 15:22:35

%S 1,11,110,1110,11010,111010,1101010,11101010,110101010,1110101010,

%T 11010101010,111010101010,1101010101010,11101010101010,

%U 110101010101010,1110101010101010,11010101010101010,111010101010101010,1101010101010101010,11101010101010101010

%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 14", 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="/A277952/b277952.txt">Table of n, a(n) for n = 0..126</a>

%H Robert Price, <a href="/A277952/a277952.tmp.txt">Diagrams of the first 20 stages</a>

%H N. J. A. Sloane, <a href="https://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 <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 06 2016: (Start)

%F G.f.: (1+10*x-x^2) / ((1-x)*(1-10*x)*(1+10*x)).

%F a(n) = a(n-1)+100*a(n-2)-100*a(n-3) for n>2.

%F a(n) = (-200-9*(-10)^n+2189*10^n)/1980. (End)

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

%t code=14; 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. A277953, A277954, A277955.

%K nonn,easy

%O 0,2

%A _Robert Price_, Nov 05 2016