A266304 - OEIS

%I #22 Apr 15 2019 06:02:06

%S 0,1,5,6,14,15,27,28,44,45,65,66,90,91,119,120,152,153,189,190,230,

%T 231,275,276,324,325,377,378,434,435,495,496,560,561,629,630,702,703,

%U 779,780,860,861,945,946,1034,1035,1127,1128,1224,1225,1325,1326,1430

%N Total number of OFF (white) cells after n iterations of the "Rule 15" 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="/A266304/b266304.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 15 2019: (Start)

%F a(n) = n*(n+(-1)^n+2)/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.: x*(1+4*x-x^2) / ((1-x)^3*(1+x)^2). (End)

%F a(n) = Sum_{i=1..n} (n-i-1) mod 2. - _Wesley Ivan Hurt_, Sep 15 2017

%t rule=15; 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 *) nwc=Table[Length[catri[[k]]]-nbc[[k]],{k,1,rows}]; (* Number of White cells in stage n *) Table[Total[Take[nwc,k]],{k,1,rows}] (* Number of White cells through stage n *)

%Y Cf. A266300.

%K nonn,easy

%O 0,3

%A _Robert Price_, Dec 26 2015