login
Total number of OFF (white) cells after n iterations of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.
2

%I #18 Apr 16 2019 15:24:33

%S 0,1,4,7,12,18,26,34,44,55,68,81,96,112,130,148,168,189,212,235,260,

%T 286,314,342,372,403,436,469,504,540,578,616,656,697,740,783,828,874,

%U 922,970,1020,1071,1124,1177,1232,1288,1346,1404,1464,1525,1588,1651

%N Total number of OFF (white) cells after n iterations of the "Rule 188" 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="/A265431/b265431.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 09 2015 and Apr 16 2019: (Start)

%F a(n) = 1/16*(10*n^2+8*n+3*(-1)^n-2*(-i)^n-2*i^n+1) where i = sqrt(-1).

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

%F (End)

%e From _Michael De Vlieger_, Dec 09 2015: (Start)

%e First 12 rows, replacing "1" with "." for better visibility of OFF cells,

%e followed by the total number of 0's per row, and the running total up to

%e that row:

%e . = 0 -> 0

%e . . 0 = 1 -> 1

%e . 0 . 0 0 = 3 -> 4

%e . . . . 0 0 0 = 3 -> 7

%e . . . 0 . 0 0 0 0 = 5 -> 12

%e . . 0 . . . 0 0 0 0 0 = 6 -> 18

%e . 0 . . . 0 . 0 0 0 0 0 0 = 8 -> 26

%e . . . . 0 . . . 0 0 0 0 0 0 0 = 8 -> 34

%e . . . 0 . . . 0 . 0 0 0 0 0 0 0 0 = 10 -> 44

%e . . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0 = 11 -> 55

%e . 0 . . . 0 . . . 0 . 0 0 0 0 0 0 0 0 0 0 = 13 -> 68

%e . . . . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0 0 0 = 13 -> 81

%e (End)

%t 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 *)

%Y Cf. A118174.

%K nonn,easy

%O 0,3

%A _Robert Price_, Dec 08 2015