A363654 Lexicographically earliest sequence of positive integers such that the n-th pair of identical terms encloses exactly a(n) terms.

1, 2, 1, 3, 2, 3, 1, 2, 4, 3, 4, 5, 3, 6, 7, 4, 3, 8, 6, 9, 7, 8, 4, 10, 3, 7, 4, 6, 8, 9, 11, 10, 12, 3, 9, 13, 7, 3, 11, 9, 14, 15, 13, 16, 17, 7, 18, 3, 19, 20, 11, 14, 9, 20, 17, 15, 19, 20, 18, 21, 11, 22, 3, 23, 24, 25, 26, 14, 17, 9, 27, 28, 29, 15, 26, 19

Offset: 1

Author

Eric Angelini and Gavin Lupo, Jun 13 2023

Keywords

• nonn

Comments

Pairs are numbered according to the position of the second term.

Examples

			a(1) = 1. The 1st constructed pair encloses 1 term:  [1, 2, 1].
a(2) = 2. The 2nd constructed pair encloses 2 terms: [2, 1, 3, 2].
a(3) = 1. The 3rd constructed pair encloses 1 term:  [3, 2, 3].
a(4) = 3. The 4th constructed pair encloses 3 terms: [1, 3, 2, 3, 1].
a(5) = 2. The 5th constructed pair encloses 2 terms: [2, 3, 1, 2].
a(6) = 3. The 6th constructed pair encloses 3 terms: [3, 1, 2, 4, 3].
a(7) = 1. The 7th constructed pair encloses 1 term:  [4, 3, 4].
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Links

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
• Eric Angelini, A seq with a strange behavior (or not)?
• Neal Gersh Tolunsky, Scatterplot of first 100000 terms

Crossrefs

Cf. A026272, A363708, A363757.

Programs

Python

from itertools import count, islice
def agen(): # generator of terms
    a, indexes = [1], {1: 0}
    yield a[-1]
    for i in count(0):
        num = 1
        while True:
            if num in indexes:
                if (len(a) - indexes[num]) == (a[i]+1):
                    an = num; indexes[an] = len(a); a.append(an); yield an
                    break
                else:
                    num += 1
            else:
                an = max(a)+1; indexes[an] = len(a); a.append(an); yield an
                num = 1
print(list(islice(agen(), 100))) # Gavin Lupo and Michael S. Branicky, Jun 13 2023