A319611 - OEIS

%I #7 Nov 09 2018 21:06:10

%S 0,0,1,1,2,2,3,2,3,4,5,5,6,6,7,6,6,7,9,8,8,10,13,11,10,11,14,15,13,14,

%T 16,13,14,16,19,17,17,17,20,20,16,18,20,19,22,21,22,21,22,20,26,29,25,

%U 26,28,28,27,27,32,30,29,30,35,31,27,28,32,32,32,34,37,30,27,36,37,39,42,41,43,41,34

%N a(n) is the number of gaps in the n-th generation of the rule-30 1D cellular automaton started from a single ON.

%C OFF cells outside the triangle of active cells are ignored.

%e The Rule-30 1D cellular automaton started from a single ON (.) cell generates the following triangle:

%e 1                          .                             a(1)= (0)

%e 2                        . . .                           a(2)= (0)

%e 3                      . . 0 0 .                         a(3)= (1)

%e 4                    . . 0 . . . .                       a(4)= (1)

%e 5                  . . 0 0 . 0 0 0 .                     a(5)= (2)

%e 6                . . 0 . . . . 0 . . .                   a(6)= (2)

%e 7              . . 0 0 . 0 0 0 0 . 0 0 .                 a(7)= (3)

%e 8            . . 0 . . . . 0 0 . . . . . .               a(8)= (2)

%e 9          . . 0 0 . 0 0 0 . . . 0 0 0 0 0 .             a(9)= (3)

%e 10       . . 0 . . . . 0 . . 0 0 . 0 0 0 . . .           a(10)=(4)

%e 11     . . 0 0 . 0 0 0 0 . 0 . . . . 0 . . 0 0 .         a(11)=(5)

%e 12   . . 0 . . . . 0 0 . . 0 . 0 0 0 0 . 0 . . . .       a(12)=(5)

%e 13 . . 0 0 . 0 0 0 . . . 0 0 . . 0 0 . . 0 . 0 0 0 .     a(13)=(6)

%t CellularAutomaton[30, {{1}, 0}, 200];

%t (Reverse[Internal`DeleteTrailingZeros[

%t       Reverse[Internal`DeleteTrailingZeros[#]]]]) & /@ %;

%t Table[

%t Length /@ Select[%[[i]] // Split, Total[#] == 0 &] // Length,

%t {i, 1, % // Length}

%t ]

%Y Cf. A100053.

%K nonn

%O 0,5

%A _Philipp O. Tsvetkov_, Sep 24 2018