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A319610 a(n) is the minimal number of successive OFF cells that appears in n-th generation of rule-30 1D cellular automaton started from a single ON cell. 2

%I

%S 0,0,2,1,2,1,2,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

%N a(n) is the minimal number of successive OFF cells that appears in n-th generation of rule-30 1D cellular automaton started from a single ON cell.

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

%H Charlie Neder, <a href="/A319610/a319610.png">Repeating pattern of length-1 runs</a>

%F G.f.: x (x + x/(1 - x) + x^3 + x^5 + x^7) (conjectured).

%F For n > 9, a(n)=1 at least up to n = 20000.

%F It is conjectured that for all n>=10, a(n)=1.

%F A period-4 pattern of length-1 runs starting at row 26 forces a(n) = 1 for all n >= 26 (see image). - _Charlie Neder_, Dec 15 2018

%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)= (2)

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

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

%e 6 . . 0 . . . . 0 . . . a(6)= (1)

%e 7 . . 0 0 . 0 0 0 0 . 0 0 . a(7)= (2)

%e 8 . . 0 . . . . 0 0 . . . . . . a(8)= (1)

%e 9 . . 0 0 . 0 0 0 . . . 0 0 0 0 0 . a(9)= (2)

%e 10 . . 0 . . . . 0 . . 0 0 . 0 0 0 . . . a(10)=(1)

%e 11 . . 0 0 . 0 0 0 0 . 0 . . . . 0 . . 0 0 . a(11)=(1)

%e 12 . . 0 . . . . 0 0 . . 0 . 0 0 0 0 . 0 . . . . a(12)=(1)

%e 13 . . 0 0 . 0 0 0 . . . 0 0 . . 0 0 . . 0 . 0 0 0 . a(13)=(1)

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

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

%t Table[Length /@ Select[%[[i]] // Split, Total[#] == 0 &] // Min, {i, 1, % // Length}]

%Y Cf. A100053.

%K nonn

%O 0,3

%A _Philipp O. Tsvetkov_, Sep 24 2018

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Last modified September 28 06:57 EDT 2021. Contains 347703 sequences. (Running on oeis4.)