A269815 - OEIS

%I #15 Feb 16 2025 08:33:30

%S 1,5,37,185,817,3425,14017,56705,228097,914945,3664897,14669825,

%T 58699777,234840065,939442177,3757932545

%N Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 35", based on the 5-celled von Neumann neighborhood.

%C Initialized with a single black (ON) cell at stage zero.

%C _Lars Blomberg_ conjectured that Rules 43 and 59 also produce this sequence.  It would be nice to have a proof.

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015

%H Eric Weisstein's World of Mathematics, <a href="https://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_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%F Conjecture: a(n) = 14*4^(n-1) - 5*2^n + 1, n>0. - _Lars Blomberg_, Apr 18 2016

%F Conjectures from _Colin Barker_, Apr 18 2016: (Start)

%F a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.

%F G.f.: (1-2*x+16*x^2-12*x^3) / ((1-x)*(1-2*x)*(1-4*x)).

%F (End)

%t rule=35; stages=300;

%t ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *)

%t on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)

%t Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)

%Y Cf. A269814.

%K nonn,more,changed

%O 0,2

%A _Robert Price_, Mar 05 2016

%E Corrected a(8) and a(9)-a(15) from _Lars Blomberg_, Apr 18 2016