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Total number of ON (black) cells after n iterations of the "Rule 47" elementary cellular automaton starting with a single ON (black) cell.
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%I #13 Apr 18 2019 10:57:25

%S 1,3,5,10,12,21,23,36,38,55,57,78,80,105,107,136,138,171,173,210,212,

%T 253,255,300,302,351,353,406,408,465,467,528,530,595,597,666,668,741,

%U 743,820,822,903,905,990,992,1081,1083,1176,1178,1275,1277,1378,1380

%N Total number of ON (black) cells after n iterations of the "Rule 47" 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="/A266663/b266663.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 03 2016 and Apr 18 2019: (Start)

%F a(n) = (n^2-(-1)^n*n+2*n+(-1)^n+3)/2 for n>0.

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

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

%F (End)

%t rule=47; 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 *) Table[Total[Take[nbc,k]],{k,1,rows}] (* Number of Black cells through stage n *)

%Y Cf. A266659.

%K nonn,easy

%O 0,2

%A _Robert Price_, Jan 02 2016