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Total number of OFF (white) cells after n iterations of the "Rule 61" elementary cellular automaton starting with a single ON (black) cell.
2

%I #16 Apr 17 2019 08:55:59

%S 0,1,4,6,11,15,23,27,39,43,59,63,83,87,111,115,143,147,179,183,219,

%T 223,263,267,311,315,363,367,419,423,479,483,543,547,611,615,683,687,

%U 759,763,839,843,923,927,1011,1015,1103,1107,1199,1203,1299,1303,1403

%N Total number of OFF (white) cells after n iterations of the "Rule 61" 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="/A266795/b266795.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 04 2016 and Apr 17 2019: (Start)

%F a(n) = (2*n*(n+(-1)^n+1)-7*(-1)^n+3)/4 for n>3.

%F a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>8.

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

%F (End)

%t rule=61; 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. A266786.

%Y Partial sums of A266794.

%K nonn,easy

%O 0,3

%A _Robert Price_, Jan 03 2016