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A266287 Total number of OFF (white) cells after n iterations of the "Rule 13" elementary cellular automaton starting with a single ON (black) cell. 1


%S 0,2,5,8,14,18,27,32,44,50,65,72,90,98,119,128,152,162,189,200,230,

%T 242,275,288,324,338,377,392,434,450,495,512,560,578,629,648,702,722,

%U 779,800,860,882,945,968,1034,1058,1127,1152,1224,1250,1325,1352,1430

%N Total number of OFF (white) cells after n iterations of the "Rule 13" elementary cellular automaton starting with a single ON (black) cell.

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

%H Robert Price, <a href="/A266287/b266287.txt">Table of n, a(n) for n = 0..999</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</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_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%F Conjectures from _Colin Barker_, Dec 28 2015 and Apr 14 2019: (Start)

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

%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.: x*(2+3*x-x^2) / ((1-x)^3*(1+x)^2).

%F (End)

%t rule=13; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) nbc=Table[Total[catri[[k]]],{k,1,rows}]; (* Number of Black cells in stage n *) nwc=Table[Length[catri[[k]]]-nbc[[k]],{k,1,rows}]; (* Number of White cells in stage n *) Table[Total[Take[nwc,k]],{k,1,rows}] (* Number of White cells through stage n *)

%Y Cf. A266282.

%K nonn,easy

%O 0,2

%A _Robert Price_, Dec 26 2015

%E Conjectures from _Colin Barker_, Apr 14 2019

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Last modified January 19 09:35 EST 2020. Contains 331048 sequences. (Running on oeis4.)