A267459 - OEIS

%I #23 Apr 17 2019 09:54:37

%S 1,2,3,4,7,10,15,20,27,34,43,52,63,74,87,100,115,130,147,164,183,202,

%T 223,244,267,290,315,340,367,394,423,452,483,514,547,580,615,650,687,

%U 724,763,802,843,884,927,970,1015,1060,1107,1154,1203,1252,1303,1354

%N Total number of ON (black) cells after n iterations of the "Rule 133" elementary cellular automaton starting with a single ON (black) cell.

%C Identical to A105343(n-1) for n > 1. - _Guenther Schrack_, Jun 01 2018

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

%H Robert Price, <a href="/A267459/b267459.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 16 2016 and Apr 17 2019: (Start)

%F a(n) = (2*n^2 - 4*n + (-1)^n + 11)/4 for n > 0.

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

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

%F (End)

%t rule=133; 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. A267423.

%K nonn,easy

%O 0,2

%A _Robert Price_, Jan 15 2016