A265430 Number of OFF (white) cells in the n-th iteration of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.
0, 1, 3, 3, 5, 6, 8, 8, 10, 11, 13, 13, 15, 16, 18, 18, 20, 21, 23, 23, 25, 26, 28, 28, 30, 31, 33, 33, 35, 36, 38, 38, 40, 41, 43, 43, 45, 46, 48, 48, 50, 51, 53, 53, 55, 56, 58, 58, 60, 61, 63, 63, 65, 66, 68, 68, 70, 71, 73, 73, 75, 76, 78, 78, 80, 81, 83
Offset: 0
Examples
From _Michael De Vlieger_, Dec 09 2015: (Start)
First 12 rows, replacing "1" with "." for better visibility of OFF cells, with total number of 0's in the row to the right of the chart:
. = 0
. . 0 = 1
. 0 . 0 0 = 3
. . . . 0 0 0 = 3
. . . 0 . 0 0 0 0 = 5
. . 0 . . . 0 0 0 0 0 = 6
. 0 . . . 0 . 0 0 0 0 0 0 = 8
. . . . 0 . . . 0 0 0 0 0 0 0 = 8
. . . 0 . . . 0 . 0 0 0 0 0 0 0 0 = 10
. . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0 = 11
. 0 . . . 0 . . . 0 . 0 0 0 0 0 0 0 0 0 0 = 13
. . . . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0 0 0 = 13
(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
Crossrefs
Cf. A118174.
Programs
-
Mathematica
lim = 134; a = {}; Do[AppendTo[a, Take[#[[k]], 2 (k - 1) + 1]], {k, Floor[Length[#]/2]}] &@ CellularAutomaton[188, {{1}, 0}, lim]; Count[#, n_ /; n == 0] & /@ a (* Michael De Vlieger, Dec 09 2015 *)
Formula
Conjectures from Colin Barker, Dec 09 2015 and Apr 16 2019: (Start)
a(n) = 1/8*(10*n+3*(-1)^n-(1-i)*(-i)^n-(1+i)*i^n-1) where i = sqrt(-1).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>4.
G.f.: x*(1+2*x+2*x^3) / ((1-x)^2*(1+x)*(1+x^2)).
(End)