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A267610 Total number of OFF (white) cells after n iterations of the "Rule 182" elementary cellular automaton starting with a single ON (black) cell.

%I A267610 #26 Feb 16 2025 08:33:29
%S A267610 0,0,2,2,4,6,12,12,14,16,22,24,30,36,50,50,52,54,60,62,68,74,88,90,96,
%T A267610 102,116,122,136,150,180,180,182,184,190,192,198,204,218,220,226,232,
%U A267610 246,252,266,280,310,312,318,324,338,344,358,372,402,408,422,436
%N A267610 Total number of OFF (white) cells after n iterations of the "Rule 182" elementary cellular automaton starting with a single ON (black) cell.
%C A267610 From _Gus Wiseman_, Mar 30 2019: (Start)
%C A267610 It appears that a(n) is also the number of increasing binary-containment pairs of distinct positive integers up to n + 1. A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second. For example, the a(2) = 2 through a(8) = 14 pairs are:
%C A267610   {1,3}  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}
%C A267610   {2,3}  {2,3}  {1,5}  {1,5}  {1,5}  {1,5}  {1,5}
%C A267610                 {2,3}  {2,3}  {1,7}  {1,7}  {1,7}
%C A267610                 {4,5}  {2,6}  {2,3}  {2,3}  {1,9}
%C A267610                        {4,5}  {2,6}  {2,6}  {2,3}
%C A267610                        {4,6}  {2,7}  {2,7}  {2,6}
%C A267610                               {3,7}  {3,7}  {2,7}
%C A267610                               {4,5}  {4,5}  {3,7}
%C A267610                               {4,6}  {4,6}  {4,5}
%C A267610                               {4,7}  {4,7}  {4,6}
%C A267610                               {5,7}  {5,7}  {4,7}
%C A267610                               {6,7}  {6,7}  {5,7}
%C A267610                                             {6,7}
%C A267610                                             {8,9}
%C A267610 (End)
%D A267610 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A267610 Robert Price, <a href="/A267610/b267610.txt">Table of n, a(n) for n = 0..1000</a>
%H A267610 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A267610 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A267610 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A267610 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A267610 Conjecture: a(n) = A267700(n) - n. - _Gus Wiseman_, Mar 30 2019
%F A267610 G.f.: (1/x)*(A(x)/x - (x+1)/(1-x)^2) where A(x) is the g.f. for A006046 (conjectured). - _John Tyler Rascoe_, Jul 08 2024
%t A267610 rule=182; 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 *) nwc=Table[Length[catri[[k]]]-nbc[[k]],{k,1,rows}]; (* Number of White cells in stage n *) Table[Total[Take[nwc,k]],{k,1,rows}] (* Number of White cells through stage n *)
%Y A267610 Cf. A000120, A006046, A006218, A070939, A071038, A080572, A267700.
%Y A267610 Cf. A325101, A325103, A325104, A325106, A325109, A325110, A325123, A325124.
%K A267610 nonn,easy
%O A267610 0,3
%A A267610 _Robert Price_, Jan 18 2016