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Total number of ON (black) cells after n iterations of the "Rule 9" elementary cellular automaton starting with a single ON (black) cell.
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%I #16 Apr 14 2019 11:53:03

%S 1,1,3,5,9,14,18,27,31,44,48,65,69,90,94,119,123,152,156,189,193,230,

%T 234,275,279,324,328,377,381,434,438,495,499,560,564,629,633,702,706,

%U 779,783,860,864,945,949,1034,1038,1127,1131,1224,1228,1325,1329,1430

%N Total number of ON (black) cells after n iterations of the "Rule 9" 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="/A266250/b266250.txt">Table of n, a(n) for n = 0..999</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</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_, Dec 28 2015 and Apr 14 2019: (Start)

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

%F a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>7.

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

%F (End)

%t rule=9; 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. A266243.

%K nonn,easy

%O 0,3

%A _Robert Price_, Dec 25 2015

%E Conjectures from _Colin Barker_, Apr 14 2019