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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.
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%I #22 Jun 29 2023 13:32:29

%S 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 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

%N 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 Charlie Neder, <a href="/A319658/a319658.png">Repeating pattern of length-1 runs</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

%F G.f.: 1/(1 - x) + 2 x + x^3 + x^5 + x^7 + x^13 (conjectured).

%F For n > 14, a(n)=1 at least until n = 10000.

%F It is conjectured that for all n >= 15, a(n)=1.

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

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

%e 2 . . . a(2)= (3)

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

%e 4 . . 0 . . . . a(4)= (2)

%e 5 . . 0 0 . 0 0 0 . a(5)= (1)

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

%e 7 . . 0 0 . 0 0 0 0 . 0 0 . a(7)= (1)

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

%e 9 . . 0 0 . 0 0 0 . . . 0 0 0 0 0 . a(9)= (1)

%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}, 100];

%t (Reverse[Internal`DeleteTrailingZeros[

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

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

%t 1, % // Length}]

%Y Cf. A319610, A319610, A100053.

%K nonn

%O 1,2

%A _Philipp O. Tsvetkov_, Sep 25 2018