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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. 3

%I #29 Apr 16 2019 17:35:05

%S 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 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 109,1,113,1,117,1,121,1,125,1,129,1,133,1,137,1,141

%N 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 This sequence is A000012 and A016813 interspersed.

%C Also column 1 of A271343. - _Omar E. Pol_, Apr 06 2016

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

%H G. C. Greubel, <a href="/A266072/b266072.txt">Table of n, a(n) for n = 0..2000</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 Conjectured g.f.: (1 + x - x^2 + 3*x^3)/(-1 + x^2)^2. - _Michael De Vlieger_, Dec 21 2015

%F Conjectures from _Colin Barker_, Dec 21 2015: (Start)

%F a(n) = n-(-1)^n*(n-1).

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

%e From _Michael De Vlieger_, Dec 21 2015: (Start)

%e First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row:

%e 1 = 1

%e 1 . . = 1

%e . . . 1 . = 1

%e 1 1 1 1 . . 1 = 5

%e . . . . . . 1 . . = 1

%e 1 1 1 1 1 1 1 . . 1 1 = 9

%e . . . . . . . . . 1 . . . = 1

%e 1 1 1 1 1 1 1 1 1 1 . . 1 1 1 = 13

%e . . . . . . . . . . . . 1 . . . . = 1

%e 1 1 1 1 1 1 1 1 1 1 1 1 1 . . 1 1 1 1 = 17

%e . . . . . . . . . . . . . . . 1 . . . . . = 1

%e 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . . 1 1 1 1 1 = 21

%e (End)

%t 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 Cf. A000012, A016813, A271343.

%K nonn,easy

%O 0,4

%A _Robert Price_, Dec 20 2015

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Last modified March 28 16:12 EDT 2024. Contains 371254 sequences. (Running on oeis4.)