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.

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, 53, 17, 16, 15, 18, 37, 25, 324, 68, 27, 42, 36, 24, 371, 91, 28, 121, 34, 59, 38, 46, 52, 73, 29, 63, 45, 72, 84, 251, 213, 48, 93, 64, 131, 57, 69

Offset: 1

Author

Eric Angelini and Carole Dubois, Dec 09 2020

Keywords

• nonn
• look
• base

Comments

This is the lexicographically earliest sequence of distinct nonnegative terms with this property.

From Edward Moody, Dec 16 2020: (Start)

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:

If F does not end with 1 or 2, no additional prefix is required, e.g. 1, 2, 13, 20, 0, 3, 12132003, 1213200312132003, ...;

If F ends with 2, but not 32, the next term can begin 131003, e.g. [...]2, 131003, 12132003, 1213200312132003, ...;

If F ends with 32, but not 2432, the next term can begin 1410014, e.g. [...]32, 1410014, 12132003, 1213200312132003, ...;

If F ends with 2432, the next term can begin 15120025, e.g. [...]2432, 15120025, 12132003, 1213200312132003, ...;

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, ...;

If F ends with 121, the next term can begin 31003, e.g. [...]121, 31003, 12132003, 1213200312132003, ...

(End)

Examples

			There is 1 digit between the first 1 [of a(1) = 1] and its closest duplicate [the 1 of a(3) = 13];
There are 2 digits between the first 2 [of a(2) = 2] and its closest duplicate [the 2 of a(4) = 20];
There are 3 digits between the first 3 [of a(3) = 13] and its closest duplicate [the 3 of a(6) = 3];
There is no digit between the first 0 [of a(4) = 20] and its closest duplicate [the 0 of a(5) = 0];
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.
		

Links

• Kevin Ryde, Table of n, a(n) for n = 1..10064
• Kevin Ryde, Perl program generating terms of the sequence

Crossrefs

Cf. A339803 (strings).