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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