A270934 - OEIS

%I #19 Feb 16 2025 08:33:32

%S 1,8,4,44,9,116,9,220,9,356,9,524,9,724,9,956,9,1220,9,1516,9,1844,9,

%T 2204,9,2596,9,3020,9,3476,9,3964,9,4484,9,5036,9,5620,9,6236,9,6884,

%U 9,7564,9,8276,9,9020,9,9796,9,10604,9,11444,9,12316,9,13220,9

%N Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 221", 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="/A270934/b270934.txt">Table of n, a(n) for n = 0..128</a>

%H Robert Price, <a href="/A270934/a270934.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>

%F Conjectures from _Colin Barker_, Mar 26 2016: (Start)

%F a(n) = (5+13*(-1)^n-4*(-1+(-1)^n)*n-4*(-1+(-1)^n)*n^2)/2 for n>2.

%F a(n) = 9 for n>2 and even.

%F a(n) = 4*n^2+4*n-4 for n>2 and odd.

%F a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>8.

%F G.f.: (1+8*x+x^2+20*x^3+8*x^5-7*x^6-4*x^7+5*x^8) / ((1-x)^3*(1+x)^3). (End)

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

%t code=221; 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 Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)

%K nonn,easy,changed

%O 0,2

%A _Robert Price_, Mar 26 2016