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A267353 Total number of ON (black) cells after n iterations of the "Rule 123" elementary cellular automaton starting with a single ON (black) cell. 1

%I

%S 1,3,6,12,15,25,28,42,45,63,66,88,91,117,120,150,153,187,190,228,231,

%T 273,276,322,325,375,378,432,435,493,496,558,561,627,630,700,703,777,

%U 780,858,861,943,946,1032,1035,1125,1128,1222,1225,1323,1326,1428,1431

%N Total number of ON (black) cells after n iterations of the "Rule 123" 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="/A267353/b267353.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 14 2016: (Start)

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

%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.: (1+2*x+x^2+2*x^3-2*x^4) / ((1-x)^3*(1+x)^2).

%F (End)

%t rule=123; 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. A267349.

%K nonn

%O 0,2

%A _Robert Price_, Jan 13 2016

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Last modified June 20 09:51 EDT 2021. Contains 345162 sequences. (Running on oeis4.)