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A319658 a(n) is the minimal number of successive ON cells that appears in n-th generation of rule-30 1D cellular automaton started from a single ON cell.

%I A319658 #22 Jun 29 2023 13:32:29
%S A319658 1,3,1,2,1,2,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A319658 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 A319658 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 A319658 a(n) is the minimal number of successive ON cells that appears in n-th generation of rule-30 1D cellular automaton started from a single ON cell.
%H A319658 Charlie Neder, <a href="/A319658/a319658.png">Repeating pattern of length-1 runs</a>
%H A319658 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F A319658 G.f.: 1/(1 - x) + 2 x + x^3 + x^5 + x^7 + x^13 (conjectured).
%F A319658 For n > 14, a(n)=1 at least until n = 10000.
%F A319658 It is conjectured that for all n >= 15, a(n)=1.
%F A319658 A period-4 pattern of length-1 runs beginning on row 19 forces a(n) = 1 for all n >= 19 (see image). - _Charlie Neder_, Dec 15 2018
%e A319658 The Rule-30 1D cellular automaton started from a single ON (.) cell generates the following triangle:
%e A319658 1                          .                             a(1)= (1)
%e A319658 2                        . . .                           a(2)= (3)
%e A319658 3                      . . 0 0 .                         a(3)= (1)
%e A319658 4                    . . 0 . . . .                       a(4)= (2)
%e A319658 5                  . . 0 0 . 0 0 0 .                     a(5)= (1)
%e A319658 6                . . 0 . . . . 0 . . .                   a(6)= (2)
%e A319658 7              . . 0 0 . 0 0 0 0 . 0 0 .                 a(7)= (1)
%e A319658 8            . . 0 . . . . 0 0 . . . . . .               a(8)= (2)
%e A319658 9          . . 0 0 . 0 0 0 . . . 0 0 0 0 0 .             a(9)= (1)
%e A319658 10       . . 0 . . . . 0 . . 0 0 . 0 0 0 . . .           a(10)=(1)
%e A319658 11     . . 0 0 . 0 0 0 0 . 0 . . . . 0 . . 0 0 .         a(11)=(1)
%e A319658 12   . . 0 . . . . 0 0 . . 0 . 0 0 0 0 . 0 . . . .       a(12)=(1)
%e A319658 13 . . 0 0 . 0 0 0 . . . 0 0 . . 0 0 . . 0 . 0 0 0 .     a(13)=(1)
%t A319658 CellularAutomaton[30, {{1}, 0}, 100];
%t A319658 (Reverse[Internal`DeleteTrailingZeros[
%t A319658       Reverse[Internal`DeleteTrailingZeros[#]]]]) & /@ %;
%t A319658 Table[Length /@ Select[%[[i]] // Split, Total[#] > 0 &] // Min, {i,
%t A319658   1, % // Length}]
%Y A319658 Cf. A319610, A319610, A100053.
%K A319658 nonn
%O A319658 1,2
%A A319658 _Philipp O. Tsvetkov_, Sep 25 2018