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A319606 a(n) is that generation of the rule-30 1D cellular automaton started from a single ON cell in which n successive OFF cells appears for the first time after a(n-1).

%I A319606 #11 Jul 06 2020 23:01:28
%S A319606 1,4,5,9,11,21,34,45,51,88,106,131,137,158,193,251,517,772,1029,1283,
%T A319606 1539,1794,2052,2305,2561,4101,5121,8197,10241,12291,16388,20482,
%U A319606 32772,36865,49154,57345,65539,262150,294913,786437,851969,1310724,1441793,1835011
%N A319606 a(n) is that generation of the rule-30 1D cellular automaton started from a single ON cell in which n successive OFF cells appears for the first time after a(n-1).
%C A319606 OFF cells outside the triangle of active cells are ignored.
%H A319606 Rémy Sigrist, <a href="/A319606/a319606.txt">C program for A319606</a>
%e A319606 The Rule-30 1D cellular automaton started from a single ON (.) cell generates the following triangle:
%e A319606 1                          .
%e A319606 2                        . . .
%e A319606 3                      . . 0 0 .
%e A319606 4                    . . 0 . . . .
%e A319606 5                  . . 0 0 . 0 0 0 .
%e A319606 6                . . 0 . . . . 0 . . .
%e A319606 7              . . 0 0 . 0 0 0 0 . 0 0 .
%e A319606 8            . . 0 . . . . 0 0 . . . . . .
%e A319606 9          . . 0 0 . 0 0 0 . . . 0 0 0 0 0 .
%e A319606 10       . . 0 . . . . 0 . . 0 0 . 0 0 0 . . .
%e A319606 11     . . 0 0 . 0 0 0 0 . 0 . . . . 0 . . 0 0 .
%e A319606 12   . . 0 . . . . 0 0 . . 0 . 0 0 0 0 . 0 . . . .
%e A319606 13 . . 0 0 . 0 0 0 . . . 0 0 . . 0 0 . . 0 . 0 0 0 .
%e A319606 0 OFF cell appears for the first time in generation (line) 1, thus a(0) = 1;
%e A319606 1 consecutive OFF cells (0) appear for the first time after line 1 in generation (line) 4, thus a(1) = 4;
%e A319606 2 consecutive OFF cells (00) appear for the first time after (line) 4 in generation (line) 5, thus a(2) = 5. [Corrected by _Rémy Sigrist_, Jul 06 2020]
%t A319606 CellularAutomaton[30, {{1}, 0}, 20000];
%t A319606 (Reverse[Internal`DeleteTrailingZeros[
%t A319606       Reverse[Internal`DeleteTrailingZeros[#]]]]) & /@ %;
%t A319606 ls = Table[
%t A319606    Max[Differences[Position[Flatten@{1, %[[n]], 1}, 1]]] - 1, {n, 1,
%t A319606     20000}];
%t A319606 res = {1};
%t A319606 Table[Position[ls, n] // Flatten, {n, 100}];
%t A319606 For[n = 1, n < 40, n++,
%t A319606 AppendTo[res, (Select[%[[n]], # > Last[res] &, 1][[1]])]]
%t A319606 res
%o A319606 (C) See Links section.
%Y A319606 Cf. A317530.
%K A319606 nonn
%O A319606 0,2
%A A319606 _Philipp O. Tsvetkov_, Sep 24 2018
%E A319606 Data corrected and more terms from _Rémy Sigrist_, Jul 06 2020