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%I #52 Sep 08 2022 08:46:14
%S 1,3,6,12,19,29,40,54,69,87,106,128,151,177,204,234,265,299,334,372,
%T 411,453,496,542,589,639,690,744,799,857,916,978,1041,1107,1174,1244,
%U 1315,1389,1464,1542,1621,1703,1786,1872,1959,2049,2140,2234,2329,2427
%N Total number of ON (black) cells after n iterations of the "Rule 155" 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="/A263511/b263511.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>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F G.f.: (1+x+2*x^3)/((1-x)^3*(1+x)). - _Vincenzo Librandi_, Jan 18 2016
%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>3. - _Vincenzo Librandi_, Jan 18 2016
%t rule=155; 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 *)
%t LinearRecurrence[{2, 0, -2, 1}, {1, 3, 6, 12}, 50] (* _Vincenzo Librandi_, Jan 18 2016 *)
%o (Magma) I:=[1,3,6,12]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..60]]; // _Vincenzo Librandi_, Jan 18 2016
%Y Cf. A263243.
%K nonn,easy
%O 0,2
%A _Robert Price_, Jan 17 2016
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