login
Number of ON (black) cells in the n-th iteration of the "Rule 37" elementary cellular automaton starting with a single ON (black) cell.
1

%I #18 Apr 18 2019 09:58:12

%S 1,1,3,2,3,6,3,10,3,14,3,18,3,22,3,26,3,30,3,34,3,38,3,42,3,46,3,50,3,

%T 54,3,58,3,62,3,66,3,70,3,74,3,78,3,82,3,86,3,90,3,94,3,98,3,102,3,

%U 106,3,110,3,114,3,118,3,122,3,126,3,130,3,134,3,138

%N Number of ON (black) cells in the n-th iteration of the "Rule 37" 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="/A266593/b266593.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 02 2016 and Apr 18 2019: (Start)

%F a(n) = (-2*(-1)^n*n+2*n+7*(-1)^n-1)/2 for n>1.

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

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

%F (End)

%t rule=37; 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 *) Table[Total[catri[[k]]],{k,1,rows}] (* Number of Black cells in stage n *)

%Y Cf. A266588.

%K nonn,easy

%O 0,3

%A _Robert Price_, Jan 01 2016