A266438 - OEIS

%I #21 Apr 15 2019 14:41:23

%S 1,4,4,11,11,22,22,37,37,56,56,79,79,106,106,137,137,172,172,211,211,

%T 254,254,301,301,352,352,407,407,466,466,529,529,596,596,667,667,742,

%U 742,821,821,904,904,991,991,1082,1082,1177,1177,1276,1276,1379,1379

%N Total number of ON (black) cells after n iterations of the "Rule 23" 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="/A266438/b266438.txt">Table of n, a(n) for n = 0..500</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 Conjectured g.f.: (-(1 + x)^(-2) + (-3 + 3*x - 2*x^2)/(-1 + x)^3)/2. - _Michael De Vlieger_, Dec 29 2015

%F Conjectures from _Colin Barker_, Dec 30 2015 and Apr 15 2019: (Start)

%F a(n) = (n^2+2*n-(-1)^n*(n+1)+3)/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 (End)

%t rule=23; 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. A266434.

%K nonn,easy

%O 0,2

%A _Robert Price_, Dec 29 2015