A363708 -id:A363708 - cp's OEIS frontend

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

Eric Angelini and Gavin Lupo, Jun 13 2023

nonn

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

			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].
...

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

Cf. A026272, A363708, A363757.

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

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.

Original entry on oeis.org

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

Neal Gersh Tolunsky, Jun 23 2023

nonn

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).

			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.

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

Cf. A363654, A363708, A026272, A330896.

A382908 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 by their average index.

Original entry on oeis.org

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

Offset: 1

Neal Gersh Tolunsky, Apr 08 2025

If two pairs have the same midpoint, the pair enclosing a shorter subsequence is considered first (in other words, the pair with the later first term and earlier second term).

Calculating terms may require backtracking, since pair numbers are not fixed until enough later terms either do or don't pair with earlier terms.

			The 1st pair (1,2,1) has average index 2 and encloses a(1) = 1 term.
The 2nd pair (2,1,3,2) has average index 3.5 and encloses a(2) = 2 distinct terms.
The 7th pair (4,1,3,2,5,2,4) has average index 10 and encloses a(7) = 4 distinct terms {1,2,3,5}.
The 8th pair (2,5,2) has average index 11 and encloses a(8) = 1 term.
Notice how the 2nd term of the 8th pair a(12) = 2 occurs earlier than the 2nd term of the 7th pair a(13) = 4. Because the average index (or center of the subsequence) is earlier in the case of the pair enclosing a(7) = 4 distinct terms, we consider it earlier than the pair enclosing a(8) = 1 term. If after setting a(12) = 2 enclosing a(8) = 1 term we had not been able to find a value to create a pair with an earlier average index to enclose a(7) = 4 distinct values, it would be necessary to backtrack to a(12) = 2 and try a different candidate.

Cf. A382911, A363757, A363708, A363654.

A382911 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 average index of the pair.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 4, 2, 3, 4, 2, 5, 1

Offset: 1

Neal Gersh Tolunsky, Apr 08 2025

If two pairs have the same midpoint, the pair enclosing a longer subsequence is considered first (in other words, the pair with the earlier first term and later second term).

Calculating terms may require backtracking, since pair numbers are not fixed until enough later terms either do or don't pair with earlier terms.

			The 1st pair (1,2,1) has average index 2 and encloses a(1) = 1 terms.
The 2nd pair (2,1,3,1,2) has average index 4 and encloses a(2) = 2 distinct terms.
The 4th pair (3,1,2,4,2,3) has average index 6.5 and encloses a(4) = 3 distinct terms.
The 5th pair (2,4,2) has average index 7 and encloses a(5) = 1 term.
Notice how the 2nd term of the 5th pair a(8) = 2 occurs earlier than the 2nd term of the 4th pair a(9) = 3. Because the average index (or center of the subsequence) is earlier in the case of the pair enclosing a(4) = 3 terms, we consider it earlier than the pair enclosing a(5) = 1 terms. If after setting a(8) = 2 enclosing a(5) = 1 terms we had not been able to find a value to create a pair with an earlier average index to enclose a(4) = 3 distinct values, it would be necessary to backtrack to a(8) = 2 and try a different candidate.

Cf. A382908, A363757, A363708, A363654.

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

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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.

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A382908 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 by their average index.

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A382911 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 average index of the pair.

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