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A266223 Total number of OFF (white) cells after n iterations of the "Rule 7" elementary cellular automaton starting with a single ON (black) cell.

%I A266223 #15 Feb 16 2025 08:33:28
%S A266223 0,1,6,6,15,15,28,28,45,45,66,66,91,91,120,120,153,153,190,190,231,
%T A266223 231,276,276,325,325,378,378,435,435,496,496,561,561,630,630,703,703,
%U A266223 780,780,861,861,946,946,1035,1035,1128,1128,1225,1225,1326,1326,1431
%N A266223 Total number of OFF (white) cells after n iterations of the "Rule 7" elementary cellular automaton starting with a single ON (black) cell.
%D A266223 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A266223 Robert Price, <a href="/A266223/b266223.txt">Table of n, a(n) for n = 0..499</a>
%H A266223 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A266223 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A266223 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A266223 Conjectures from _Colin Barker_, Dec 26 2015 and Apr 14 2019: (Start)
%F A266223 a(n) = 1/2*(n+1)*(n+(-1)^n+1) for n>0.
%F A266223 a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5.
%F A266223 G.f.: x*(1+5*x-2*x^2-x^3+x^4) / ((1-x)^3*(1+x)^2).
%F A266223 (End)
%t A266223 rule=7; 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 A266223 Cf. A266216.
%K A266223 nonn,easy
%O A266223 0,3
%A A266223 _Robert Price_, Dec 24 2015
%E A266223 Conjectures from _Colin Barker_, Apr 14 2019