A266221 - OEIS

%I #18 Apr 13 2019 22:10:28

%S 1,3,3,10,10,21,21,36,36,55,55,78,78,105,105,136,136,171,171,210,210,

%T 253,253,300,300,351,351,406,406,465,465,528,528,595,595,666,666,741,

%U 741,820,820,903,903,990,990,1081,1081,1176,1176,1275,1275,1378,1378

%N Total number of ON (black) cells after n iterations 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="/A266221/b266221.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 25 2015 and Apr 13 2019: (Start)

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

%F a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5.

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

%F (End)

%F a(n) = A000217(A052928(n+1)) for n>0. - _Michel Marcus_, Sep 25 2016

%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[Total[Take[nbc,k]],{k,1,rows}] (* Number of Black cells through stage n *)

%Y Cf. A266216.

%K nonn,easy

%O 0,2

%A _Robert Price_, Dec 24 2015

%E Conjectures from _Colin Barker_, Apr 13 2019