A270008 - OEIS

%I #11 Jul 26 2024 21:16:32

%S 1,9,9,58,58,179,179,404,404,765,765,1294,1294,2023,2023,2984,2984,

%T 4209,4209,5730,5730,7579,7579,9788,9788,12389,12389,15414,15414,

%U 18895,18895,22864,22864,27353,27353,32394,32394,38019,38019,44260,44260,51149,51149

%N Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 5", 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="/A270008/b270008.txt">Table of n, a(n) for n = 0..128</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_, Mar 09 2016: (Start)

%F a(n) = (9-9*(-1)^n+(22-24*(-1)^n)*n-12*(-2+(-1)^n)*n^2+8*n^3)/12 for n>0.

%F a(n) = (4*n^3+6*n^2-n)/6 for n>0 and even.

%F a(n) = (4*n^3+18*n^2+23*n+9)/6 for n>0 and odd.

%F a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7).

%F G.f.: (1+8*x-3*x^2+25*x^3+3*x^4-2*x^5-x^6+x^7) / ((1-x)^4*(1+x)^3).

%F (End)

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

%t code=5; 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 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)

%t Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)

%Y Cf. A270006.

%K nonn,easy

%O 0,2

%A _Robert Price_, Mar 08 2016