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A265723 Number of OFF (white) cells in the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

%I A265723 #21 Feb 16 2025 08:33:28
%S A265723 0,3,4,3,8,3,12,3,16,3,20,3,24,3,28,3,32,3,36,3,40,3,44,3,48,3,52,3,
%T A265723 56,3,60,3,64,3,68,3,72,3,76,3,80,3,84,3,88,3,92,3,96,3,100,3,104,3,
%U A265723 108,3,112,3,116,3,120,3,124,3,128,3,132,3,136,3,140,3
%N A265723 Number of OFF (white) cells in the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.
%D A265723 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A265723 Robert Price, <a href="/A265723/b265723.txt">Table of n, a(n) for n = 0..999</a>
%H A265723 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A265723 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A265723 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A265723 Conjectures from _Colin Barker_, Dec 15 2015 and Apr 16 2019: (Start)
%F A265723 a(n) = 1/2*((2*(-1)^n+2)*n-3*((-1)^n-1)).
%F A265723 a(n) = 2*a(n-2) - a(n-4) for n>3.
%F A265723 G.f.: x*(3+4*x-3*x^2) / ((1-x)^2*(1+x)^2).
%F A265723 (End)
%e A265723 From _Michael De Vlieger_, Dec 14 2015: (Start)
%e A265723 First 12 rows, replacing ones with "." for better visibility of OFF cells, followed by the total number of 0's per row at right:
%e A265723                       .                        =  0
%e A265723                     0 0 0                      =  3
%e A265723                   0 0 . 0 0                    =  4
%e A265723                 . . 0 0 0 . .                  =  3
%e A265723               0 0 0 0 . 0 0 0 0                =  8
%e A265723             . . . . 0 0 0 . . . .              =  3
%e A265723           0 0 0 0 0 0 . 0 0 0 0 0 0            = 12
%e A265723         . . . . . . 0 0 0 . . . . . .          =  3
%e A265723       0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0        = 16
%e A265723     . . . . . . . . 0 0 0 . . . . . . . .      =  3
%e A265723   0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0    = 20
%e A265723 . . . . . . . . . . 0 0 0 . . . . . . . . . .  =  3
%e A265723 (End)
%t A265723 rows = 71; Count[#, n_ /; n == 0] & /@ Table[Table[Take[CellularAutomaton[1, {{1}, 0}, rows - 1, {All, All}][[k]], {rows - k + 1, rows + k - 1}], {k, 1, rows}][[k]], {k, 1, rows}] (* _Michael De Vlieger_, Dec 14 2015 *)
%Y A265723 Cf. A265718, A265720, A265721.
%K A265723 nonn,easy
%O A265723 0,2
%A A265723 _Robert Price_, Dec 14 2015