A265431 Total number of OFF (white) cells after n iterations of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.
0, 1, 4, 7, 12, 18, 26, 34, 44, 55, 68, 81, 96, 112, 130, 148, 168, 189, 212, 235, 260, 286, 314, 342, 372, 403, 436, 469, 504, 540, 578, 616, 656, 697, 740, 783, 828, 874, 922, 970, 1020, 1071, 1124, 1177, 1232, 1288, 1346, 1404, 1464, 1525, 1588, 1651
Offset: 0
Examples
From _Michael De Vlieger_, Dec 09 2015: (Start)
First 12 rows, replacing "1" with "." for better visibility of OFF cells,
followed by the total number of 0's per row, and the running total up to
that row:
. = 0 -> 0
. . 0 = 1 -> 1
. 0 . 0 0 = 3 -> 4
. . . . 0 0 0 = 3 -> 7
. . . 0 . 0 0 0 0 = 5 -> 12
. . 0 . . . 0 0 0 0 0 = 6 -> 18
. 0 . . . 0 . 0 0 0 0 0 0 = 8 -> 26
. . . . 0 . . . 0 0 0 0 0 0 0 = 8 -> 34
. . . 0 . . . 0 . 0 0 0 0 0 0 0 0 = 10 -> 44
. . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0 = 11 -> 55
. 0 . . . 0 . . . 0 . 0 0 0 0 0 0 0 0 0 0 = 13 -> 68
. . . . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0 0 0 = 13 -> 81
(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 = 104; a = {}; Do[AppendTo[a, Take[#[[k]], 2 (k - 1) + 1]], {k, Floor[Length[#]/2]}] &@ CellularAutomaton[188, {{1}, 0}, lim]; Accumulate[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) = 1/16*(10*n^2+8*n+3*(-1)^n-2*(-i)^n-2*i^n+1) where i = sqrt(-1).
G.f.: x*(1+2*x+2*x^3) / ((1-x)^3*(1+x)*(1+x^2)).
(End)