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A266072 Number of ON (black) cells in the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.

%I A266072 #30 Feb 16 2025 08:33:28
%S A266072 1,1,1,5,1,9,1,13,1,17,1,21,1,25,1,29,1,33,1,37,1,41,1,45,1,49,1,53,1,
%T A266072 57,1,61,1,65,1,69,1,73,1,77,1,81,1,85,1,89,1,93,1,97,1,101,1,105,1,
%U A266072 109,1,113,1,117,1,121,1,125,1,129,1,133,1,137,1,141
%N A266072 Number of ON (black) cells in the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.
%C A266072 This sequence is A000012 and A016813 interspersed.
%C A266072 Also column 1 of A271343. - _Omar E. Pol_, Apr 06 2016
%D A266072 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A266072 G. C. Greubel, <a href="/A266072/b266072.txt">Table of n, a(n) for n = 0..2000</a>
%H A266072 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A266072 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A266072 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A266072 Conjectured g.f.: (1 + x - x^2 + 3*x^3)/(-1 + x^2)^2. - _Michael De Vlieger_, Dec 21 2015
%F A266072 Conjectures from _Colin Barker_, Dec 21 2015: (Start)
%F A266072 a(n) = n-(-1)^n*(n-1).
%F A266072 a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
%e A266072 From _Michael De Vlieger_, Dec 21 2015: (Start)
%e A266072 First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row:
%e A266072                       1                        =  1
%e A266072                     1 . .                      =  1
%e A266072                   . . . 1 .                    =  1
%e A266072                 1 1 1 1 . . 1                  =  5
%e A266072               . . . . . . 1 . .                =  1
%e A266072             1 1 1 1 1 1 1 . . 1 1              =  9
%e A266072           . . . . . . . . . 1 . . .            =  1
%e A266072         1 1 1 1 1 1 1 1 1 1 . . 1 1 1          = 13
%e A266072       . . . . . . . . . . . . 1 . . . .        =  1
%e A266072     1 1 1 1 1 1 1 1 1 1 1 1 1 . . 1 1 1 1      = 17
%e A266072   . . . . . . . . . . . . . . . 1 . . . . .    =  1
%e A266072 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . . 1 1 1 1 1  = 21
%e A266072 (End)
%t A266072 rule = 3; rows = 30; Table[Total[Table[Take[CellularAutomaton[rule, {{1},0},rows-1,{All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}]
%Y A266072 Cf. A000012, A016813, A271343.
%K A266072 nonn,easy
%O A266072 0,4
%A A266072 _Robert Price_, Dec 20 2015