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Total number of OFF (white) cells after n iterations of the "Rule 109" elementary cellular automaton starting with a single ON (black) cell.
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%I #15 Apr 19 2019 10:06:59

%S 0,2,2,7,9,13,16,26,30,37,45,57,64,76,86,101,113,127,140,160,174,191,

%T 209,231,248,270,290,315,337,361,384,414,438,465,493,525,552,584,614,

%U 649,681,715,748,788,822,859,897,939,976,1018,1058,1103,1145,1189

%N Total number of OFF (white) cells after n iterations of the "Rule 109" 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="/A267214/b267214.txt">Table of n, a(n) for n = 0..1000</a>

%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_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%F Conjectures from _Colin Barker_, Jan 13 2016 and Apr 19 2019: (Start)

%F a(n) = a(n-1)+a(n-3)-a(n-5)-a(n-7)+a(n-8) for n>7.

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

%F (End)

%t rule=109; 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. A243566.

%K nonn,easy

%O 0,2

%A _Robert Price_, Jan 11 2016