A277954 - OEIS

%I #13 Feb 16 2025 08:33:37

%S 1,3,6,14,26,58,106,234,426,938,1706,3754,6826,15018,27306,60074,

%T 109226,240298,436906,961194,1747626,3844778,6990506,15379114,

%U 27962026,61516458,111848106,246065834,447392426,984263338,1789569706,3937053354,7158278826

%N Decimal 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="/A277954/b277954.txt">Table of n, a(n) for n = 0..126</a>

%H Robert Price, <a href="/A277954/a277954.tmp.txt">Diagrams of the 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 <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>

%H Robert Price, <a href="/A277954/a277954.tmp.txt">Diagrams of the first 20 stages</a>

%F Conjectures from _Colin Barker_, Nov 06 2016: (Start)

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

%F a(n) = a(n-1)+4*a(n-2)-4*a(n-3) for n>2.

%F a(n) = (-8-(-2)^n+21*2^n)/12. (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]],2], {i,1,stages-1}]

%t LinearRecurrence[{1, 4, -4}, {1, 3, 6}, 31] (* or *)

%t CoefficientList[ Series[(1 + 2x - x^2)/(1 - x - 4x^2 + 4x^3), {x, 0, 31}], x] (* _Robert G. Wilson v_, Nov 05 2016 *)

%Y Cf. A277952, A277953, A277955.

%K nonn,easy,changed

%O 0,2

%A _Robert Price_, Nov 05 2016