cp's OEIS Frontend

A169699 Total number of ON cells at stage n of two-dimensional 5-neighbor outer totalistic cellular automaton defined by "Rule 510".

%I A169699 #29 Feb 16 2025 08:33:11
%S A169699 1,5,12,25,28,56,56,113,60,120,120,240,120,240,240,481,124,248,248,
%T A169699 496,248,496,496,992,248,496,496,992,496,992,992,1985,252,504,504,
%U A169699 1008,504,1008,1008,2016,504,1008,1008,2016,1008,2016,2016,4032,504,1008,1008,2016
%N A169699 Total number of ON cells at stage n of two-dimensional 5-neighbor outer totalistic cellular automaton defined by "Rule 510".
%C A169699 We work on the square grid. Each cell has 4 neighbors, N, S, E, W. If none of your 4 neighbors are ON, your state does not change. If all 4 of your neighbors are ON, your state flips. In all other cases you turn ON. We start with one ON cell.
%C A169699 As observed by Packard and Wolfram (see Fig. 2), a slice along the E-W line shows the successive states of the 1-D CA Rule 126 (see A071035, A071051).
%D A169699 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A169699 Robert Price, <a href="/A169699/b169699.txt">Table of n, a(n) for n = 0..128</a>
%H A169699 N. H. Packard and S. Wolfram, <a href="http://new.math.uiuc.edu/im2008/dakkak/papers/files/wolfram.2dca.pdf">Two-Dimensional Cellular Automata</a>, Journal of Statistical Physics, 38 (1985), 901-946.
%H A169699 N. J. A. Sloane, <a href="/A169699/a169699.pdf">Illustration of first 28 generations</a>
%H A169699 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, 2015
%H A169699 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A169699 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A169699 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A169699 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A169699 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A169699 For n>0, it is easy to show that if 2^k <= n < 2^(k+1) then a(n) =
%F A169699 (2^(k+1)-1)*2^(1+wt(n)), where wt is the binary weight A000120, except that if n is a power of 2 we must add 1 to the result.
%e A169699 When arranged into blocks of sizes 1,1,2,4,8,16,...:
%e A169699 1,
%e A169699 5,
%e A169699 12, 25,
%e A169699 28, 56, 56, 113,
%e A169699 60, 120, 120, 240, 120, 240, 240, 481,
%e A169699 124, 248, 248, 496, 248, 496, 496, 992, 248, 496, 496, 992, 496, 992, 992, 1985,
%e A169699 252, 504, 504, 1008, 504, 1008, 1008, 2016, 504, 1008, 1008, 2016, 1008, 2016, 2016, 4032, 504, 1008, 1008, 2016, 1008, 2016, 2016, 4032,
%e A169699 ..., the initial terms in the rows (after the initial rows) have the form 2^m-4 and the final terms are given by A092440. The row beginning with 2^m-4 is divisible by 2^(m-2)-1 (see formula).
%p A169699 A000120 := proc(n) add(i,i=convert(n,base,2)) end:
%p A169699 ht:=n->floor(log[2](n));
%p A169699 f:=proc(n) local a,t1;
%p A169699 if n=0 then 1 else
%p A169699 a:=(2^(ht(n)+1)-1)*2^(1+A000120(n));
%p A169699 if 2^log[2](n)=n then a:=a+1; fi; a; fi; end;
%p A169699 [seq(f(n),n=0..65)]; # A169699
%t A169699 Map[Function[Apply[Plus,Flatten[ #1]]], CellularAutomaton[{ 510, {2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},100]]
%t A169699 ArrayPlot /@ CellularAutomaton[{510, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 28]
%Y A169699 Cf. A000120, A071035, A071051, A008574, A092440, A169700.
%Y A169699 See A253089 for 9-celled neighborhood version.
%K A169699 nonn,tabf
%O A169699 0,2
%A A169699 _N. J. A. Sloane_, Apr 17 2010
%E A169699 Entry revised with more precise definition, formula and additional information, _N. J. A. Sloane_, Aug 24 2014