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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. 2

%I

%S 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 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 108,3,112,3,116,3,120,3,124,3,128,3,132,3,136,3,140,3

%N 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 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

%H Robert Price, <a href="/A265723/b265723.txt">Table of n, a(n) for n = 0..999</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</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_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%F Conjectures from _Colin Barker_, Dec 15 2015 and Apr 16 2019: (Start)

%F a(n) = 1/2*((2*(-1)^n+2)*n-3*((-1)^n-1)).

%F a(n) = 2*a(n-2) - a(n-4) for n>3.

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

%F (End)

%e From _Michael De Vlieger_, Dec 14 2015: (Start)

%e 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 . = 0

%e 0 0 0 = 3

%e 0 0 . 0 0 = 4

%e . . 0 0 0 . . = 3

%e 0 0 0 0 . 0 0 0 0 = 8

%e . . . . 0 0 0 . . . . = 3

%e 0 0 0 0 0 0 . 0 0 0 0 0 0 = 12

%e . . . . . . 0 0 0 . . . . . . = 3

%e 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 = 16

%e . . . . . . . . 0 0 0 . . . . . . . . = 3

%e 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 = 20

%e . . . . . . . . . . 0 0 0 . . . . . . . . . . = 3

%e (End)

%t 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 Cf. A265718, A265720, A265721.

%K nonn,easy

%O 0,2

%A _Robert Price_, Dec 14 2015

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Last modified July 29 06:36 EDT 2021. Contains 346340 sequences. (Running on oeis4.)