A363757 Lexicographically earliest sequence of positive integers such that the n-th pair of consecutive equal values are separated by a(n) distinct terms, with pairs numbered according to the position of the second term in the pair.

1, 2, 1, 3, 2, 3, 4, 1, 3, 2, 5, 4, 5, 3, 4, 6, 1, 5, 2, 6, 4, 7, 3, 7, 5, 3, 1, 4, 8, 2, 1, 6, 3, 2, 3, 8, 9, 7, 8, 7, 1, 9, 7, 8, 5, 10, 4, 3, 2, 9, 2, 6, 8, 7, 3, 11, 1, 8, 3, 1, 10, 3, 6, 9, 7, 3, 12, 5, 12, 8, 3, 8, 2, 12, 9, 1, 7, 12, 13, 4, 9, 11, 8, 4, 2, 8, 10, 1, 10, 13, 6

Offset: 1

Author

Neal Gersh Tolunsky, Jun 23 2023

Keywords

• nonn

Comments

The word 'distinct' differentiates this sequence from A363654.

A000124 gives the index of the first occurrence of n, and A080036 gives the indices of the remaining terms. A record high term occurs when its corresponding pair number would be the previous record high, since that would have to use all terms between the enclosing pair, which is impossible.

A083920(n) gives the number of pairs in the first n terms of this sequence.

If pairs are numbered according to the position of the first term in the pair (rather than second), this becomes A001511 (the ruler function).

Examples

			The 1st pair (1,2,1) encloses 1 term because a(1)=1.
The 2nd pair (2,1,3,2) encloses 2 distinct terms because a(2)=2.
The 3rd pair (3,2,3) encloses 1 term because a(3)=1.
The 4th pair (1,3,2,3,4,1) encloses 3 distinct terms because a(4)=3.
a(4)=3 since if we place a 1 or a 2 (creating the second pair), this would enclose less than a(2)=2 distinct terms, so a(4) must be the smallest unused number, which is 3.
		

Links

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
• Neal Gersh Tolunsky, Graph of first 100000 terms

Crossrefs

Cf. A363654, A363708, A026272, A330896.