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A317530 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.

%I A317530 #24 Jul 08 2020 20:44:31
%S A317530 4,3,5,7,9,17,45,33,44,67,47,66,130,65,129,517,260,516,259,515,258,
%T A317530 514,257,513,4101,1025,4100,2049,4099,16388,4098,16387,4097,16386,
%U A317530 8193,16385,262150,32769,262149,65537,262148,131073,262147,524291,262146,524290,262145,524289
%N A317530 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.
%C A317530 OFF cells outside the triangle of active cells are ignored.
%C A317530 This is also the position at which n appears for the first time in A100053.
%H A317530 Rémy Sigrist, <a href="/A317530/a317530.txt">C program for A317530</a>
%e A317530 The Rule-30 1D cellular automaton started from a single ON (.) cell generates the following triangle:
%e A317530 1                          .
%e A317530 2                        . . .
%e A317530 3                      . . 0 0 .
%e A317530 4                    . . 0 . . . .
%e A317530 5                  . . 0 0 . 0 0 0 .
%e A317530 6                . . 0 . . . . 0 . . .
%e A317530 7              . . 0 0 . 0 0 0 0 . 0 0 .
%e A317530 8            . . 0 . . . . 0 0 . . . . . .
%e A317530 9          . . 0 0 . 0 0 0 . . . 0 0 0 0 0 .
%e A317530 10       . . 0 . . . . 0 . . 0 0 . 0 0 0 . . .
%e A317530 11     . . 0 0 . 0 0 0 0 . 0 . . . . 0 . . 0 0 .
%e A317530 12   . . 0 . . . . 0 0 . . 0 . 0 0 0 0 . 0 . . . .
%e A317530 13 . . 0 0 . 0 0 0 . . . 0 0 . . 0 0 . . 0 . 0 0 0 .
%e A317530 1 OFF cell (0) appears for the first time in generation (line) 4, thus a(1) = 4;
%e A317530 2 consecutive OFF cells (00) appear for the first time in generation (line) 3, thus a(2) = 3;
%e A317530 5 consecutive OFF cells (00000) appear for the first time in generation (line) 9, thus a(5) = 9;
%e A317530 ...
%t A317530 CellularAutomaton[30, {{1}, 0}, 10000];
%t A317530 (Reverse[Internal`DeleteTrailingZeros[
%t A317530       Reverse[Internal`DeleteTrailingZeros[#]]]]) & /@ %;
%t A317530 Table[Position[(Max@FoldList[If[#2 == 0, #1 + 1, 0] &, 0, #]) & /@ %, i] //
%t A317530    Flatten // First, {i, 1, 29}]
%o A317530 (C) See Links section.
%Y A317530 Cf. A100053.
%K A317530 nonn
%O A317530 1,1
%A A317530 _Philipp O. Tsvetkov_, Jul 30 2018
%E A317530 a(30)-a(36) from _Robert G. Wilson v_, Aug 07 2018
%E A317530 Data corrected and more terms from _Rémy Sigrist_, Jul 06 2020