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.
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, 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, 108, 3, 112, 3, 116, 3, 120, 3, 124, 3, 128, 3, 132, 3, 136, 3, 140, 3
Offset: 0
Examples
From _Michael De Vlieger_, Dec 14 2015: (Start)
First 12 rows, replacing ones with "." for better visibility of OFF cells, followed by the total number of 0's per row at right:
. = 0
0 0 0 = 3
0 0 . 0 0 = 4
. . 0 0 0 . . = 3
0 0 0 0 . 0 0 0 0 = 8
. . . . 0 0 0 . . . . = 3
0 0 0 0 0 0 . 0 0 0 0 0 0 = 12
. . . . . . 0 0 0 . . . . . . = 3
0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 = 16
. . . . . . . . 0 0 0 . . . . . . . . = 3
0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 = 20
. . . . . . . . . . 0 0 0 . . . . . . . . . . = 3
(End)
References
- S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
Links
- Robert Price, Table of n, a(n) for n = 0..999
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- Index entries for sequences related to cellular automata
- Index to Elementary Cellular Automata
Programs
-
Mathematica
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 *)
Formula
Conjectures from Colin Barker, Dec 15 2015 and Apr 16 2019: (Start)
a(n) = 1/2*((2*(-1)^n+2)*n-3*((-1)^n-1)).
a(n) = 2*a(n-2) - a(n-4) for n>3.
G.f.: x*(3+4*x-3*x^2) / ((1-x)^2*(1+x)^2).
(End)