A378876 a(1)=1; thereafter a(n) is the smallest k for which the subsequence a(n-k..n-1) has a distinct multiset from that of any other subsequence of the sequence thus far.

1, 1, 2, 1, 4, 1, 3, 1, 3, 3, 2, 2, 2, 3, 6, 1, 2, 3, 3, 6, 3, 4, 2, 2, 3, 5, 1, 2, 3, 5, 5, 2, 2, 3, 6, 5, 2, 3, 5, 5, 5, 3, 5, 5, 6, 3, 5, 9, 1, 2, 3, 4, 4, 2, 3, 5, 4, 2, 3, 5, 5, 6, 6, 2, 2, 3, 5, 8, 1, 2, 3, 4, 5, 5, 3, 5, 5, 6, 7, 1, 2, 3, 4, 5, 6, 3, 5, 5

Offset: 1

Author

Neal Gersh Tolunsky, Jan 17 2025

Keywords

• nonn

Comments

In other words, a(n) is the length of the shortest subsequence ending at a(n-1) which has a unique multiset among all multisets of subsequences of the sequence thus far. Alternatively, this is (the length of the longest subsequence ending at a(n-1) whose multiset has occurred before as that of another subsequence) plus 1.

Examples

			a(15) = 6 because the length-6 subsequence a(9..14) =  3,3,2,2,2,3 has the shortest unique multiset, which does not occur elsewhere as the multiset of any other subsequence in the sequence thus far. No shorter subsequence ending in a(14) with a unique ordinal transform exists in the sequence thus far. For example, a(15) cannot be 5 because the length-5 subsequence a(10..14) = 3,2,2,2,3 has the same multiset as that of the subsequence a(9..13) = 3,3,2,2,2.
		

Crossrefs

Cf. A376937, A375207, A377079.