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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.

%I A319610 #27 Jun 29 2023 13:30:14
%S A319610 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 A319610 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 A319610 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 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.
%C A319610 OFF cells outside the triangle of active cells are ignored.
%H A319610 Charlie Neder, <a href="/A319610/a319610.png">Repeating pattern of length-1 runs</a>
%H A319610 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F A319610 G.f.: x (x + x/(1 - x) + x^3 + x^5 + x^7) (conjectured).
%F A319610 For n > 9, a(n)=1 at least up to n = 20000.
%F A319610 It is conjectured that for all n>=10, a(n)=1.
%F A319610 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 A319610 The Rule-30 1D cellular automaton started from a single ON (.) cell generates the following triangle:
%e A319610 1                          .                             a(1)= (0)
%e A319610 2                        . . .                           a(2)= (0)
%e A319610 3                      . . 0 0 .                         a(3)= (2)
%e A319610 4                    . . 0 . . . .                       a(4)= (1)
%e A319610 5                  . . 0 0 . 0 0 0 .                     a(5)= (2)
%e A319610 6                . . 0 . . . . 0 . . .                   a(6)= (1)
%e A319610 7              . . 0 0 . 0 0 0 0 . 0 0 .                 a(7)= (2)
%e A319610 8            . . 0 . . . . 0 0 . . . . . .               a(8)= (1)
%e A319610 9          . . 0 0 . 0 0 0 . . . 0 0 0 0 0 .             a(9)= (2)
%e A319610 10       . . 0 . . . . 0 . . 0 0 . 0 0 0 . . .           a(10)=(1)
%e A319610 11     . . 0 0 . 0 0 0 0 . 0 . . . . 0 . . 0 0 .         a(11)=(1)
%e A319610 12   . . 0 . . . . 0 0 . . 0 . 0 0 0 0 . 0 . . . .       a(12)=(1)
%e A319610 13 . . 0 0 . 0 0 0 . . . 0 0 . . 0 0 . . 0 . 0 0 0 .     a(13)=(1)
%t A319610 CellularAutomaton[30, {{1}, 0}, 200];
%t A319610 (Reverse[Internal`DeleteTrailingZeros[Reverse[Internal`DeleteTrailingZeros[#]]]]) & /@ %;
%t A319610 Table[Length /@ Select[%[[i]] // Split, Total[#] == 0 &] // Min, {i, 1, % // Length}]
%Y A319610 Cf. A100053.
%K A319610 nonn
%O A319610 0,3
%A A319610 _Philipp O. Tsvetkov_, Sep 24 2018