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A339611 Pick any digit d; there are exactly d digits between d and the closest duplicate of d (either before or after) in the sequence.

%I A339611 #51 Feb 01 2021 13:40:00
%S A339611 1,2,13,20,0,3,4,5,6,7,8,41,51,61,71,81,9,12,14,21,31,49,32,54,23,62,
%T A339611 53,17,16,15,18,37,25,324,68,27,42,36,24,371,91,28,121,34,59,38,46,52,
%U A339611 73,29,63,45,72,84,251,213,48,93,64,131,57,69
%N A339611 Pick any digit d; there are exactly d digits between d and the closest duplicate of d (either before or after) in the sequence.
%C A339611 This is the lexicographically earliest sequence of distinct nonnegative terms with this property.
%C A339611 From _Edward Moody_, Dec 16 2020: (Start)
%C A339611 The sequence is infinite because any potential earliest finite sequence F with the property (i.e., with no "unmatched" digits) can be extended to an infinite sequence. One way to do this is by adding terms consisting of sufficient copies of the string 12132003 to avoid duplication. In some cases the first additional term needs an additional prefix to preserve the property:
%C A339611 If F does not end with 1 or 2, no additional prefix is required, e.g. 1, 2, 13, 20, 0, 3, 12132003, 1213200312132003, ...;
%C A339611 If F ends with 2, but not 32, the next term can begin 131003, e.g. [...]2, 131003, 12132003, 1213200312132003, ...;
%C A339611 If F ends with 32, but not 2432, the next term can begin 1410014, e.g. [...]32, 1410014, 12132003, 1213200312132003, ...;
%C A339611 If F ends with 2432, the next term can begin 15120025, e.g. [...]2432, 15120025, 12132003, 1213200312132003, ...;
%C A339611 If F ends with 1, but not 121, the next term can begin 2132003, e.g. 1, 2, 13, 20, 0, 3, 4, 5, 6, 7, 8, 41, 51, 61, 71, 81, 2132003, 12132003, 1213200312132003, ...;
%C A339611 If F ends with 121, the next term can begin 31003, e.g. [...]121, 31003, 12132003, 1213200312132003, ...
%C A339611 (End)
%H A339611 Kevin Ryde, <a href="/A339611/b339611.txt">Table of n, a(n) for n = 1..10064</a>
%H A339611 Kevin Ryde, <a href="/A339611/a339611.pl.txt">Perl program generating terms of the sequence</a>
%e A339611 There is 1 digit between the first 1 [of a(1) = 1] and its closest duplicate [the 1 of a(3) = 13];
%e A339611 There are 2 digits between the first 2 [of a(2) = 2] and its closest duplicate [the 2 of a(4) = 20];
%e A339611 There are 3 digits between the first 3 [of a(3) = 13] and its closest duplicate [the 3 of a(6) = 3];
%e A339611 There is no digit between the first 0 [of a(4) = 20] and its closest duplicate [the 0 of a(5) = 0];
%e A339611 There is no contradiction when picking the 1 of a(13) = 51 as there is 1 digit between it and the 1 of a(12) = 41 and 1 digit between it and the 1 of a(14) = 61; etc.
%Y A339611 Cf. A339803 (strings).
%K A339611 nonn,look,base
%O A339611 1,2
%A A339611 _Eric Angelini_ and _Carole Dubois_, Dec 09 2020