A266222 - OEIS

%I #16 Apr 13 2019 22:10:35

%S 0,1,5,0,9,0,13,0,17,0,21,0,25,0,29,0,33,0,37,0,41,0,45,0,49,0,53,0,

%T 57,0,61,0,65,0,69,0,73,0,77,0,81,0,85,0,89,0,93,0,97,0,101,0,105,0,

%U 109,0,113,0,117,0,121,0,125,0,129,0,133,0,137,0,141,0

%N Number of OFF (white) cells in the n-th iteration of the "Rule 7" 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="/A266222/b266222.txt">Table of n, a(n) for n = 0..499</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 26 2015 and Apr 13 2019: (Start)

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

%F a(n) = 2*a(n-2) - a(n-4) for n>5.

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

%F (End)

%t rule=7; 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[Length[catri[[k]]]-nbc[[k]],{k,1,rows}] (* Number of White cells in stage n *)

%Y Cf. A266216.

%K nonn,easy

%O 0,3

%A _Robert Price_, Dec 24 2015