A270681 - OEIS

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

%S 1,5,12,24,32,52,60,88,96,132,140,184,192,244,252,312,320,388,396,472,

%T 480,564,572,664,672,772,780,888,896,1012,1020,1144,1152,1284,1292,

%U 1432,1440,1588,1596,1752,1760,1924,1932,2104,2112,2292,2300,2488,2496,2692

%N Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 190", 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="/A270681/b270681.txt">Table of n, a(n) for n = 0..128</a>

%H Robert Price, <a href="/A270681/a270681.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 21 2016: (Start)

%F a(n) = 3/2*(-1+(-1)^n)-(-5+(-1)^n)*n+n^2 for n>1.

%F a(n) = n^2+4*n for n>1 and even.

%F a(n) = n^2+6*n-3 for n>1 and odd.

%F a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>4.

%F G.f.: (1+4*x+5*x^2+4*x^3-5*x^4-x^6) / ((1-x)^3*(1+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=190; 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

%O 0,2

%A _Robert Price_, Mar 21 2016