A272567 - OEIS

%I #11 Jul 26 2024 21:16:40

%S 7,-3,39,-31,99,-87,183,-171,303,-291,455,-443,639,-627,855,-843,1103,

%T -1091,1383,-1371,1695,-1683,2039,-2027,2415,-2403,2823,-2811,3263,

%U -3251,3735,-3723,4239,-4227,4775,-4763,5343,-5331,5943,-5931,6575,-6563,7239

%N First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 501", 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="/A272567/b272567.txt">Table of n, a(n) for n = 0..127</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_, May 03 2016: (Start)

%F a(n) = 6+9*(-1)^n+4*n+4*(-1)^n*n^2 for n>5.

%F a(n) = 4*n^2+4*n+15 for n>5 and even.

%F a(n) = -4*n^2+4*n-3 for n>5 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.: (7+4*x+22*x^2+3*x^4-4*x^6-4*x^7+8*x^8-4*x^10) / ((1-x)^2*(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=501; 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[on[[i+1]]-on[[i]],{i,1,Length[on]-1}] (* Difference at each stage *)

%Y Cf. A272564.

%K sign,easy

%O 0,1

%A _Robert Price_, May 02 2016