A366691
Lexicographically earliest sequence such that each set of terms enclosed by two equal values, excluding the endpoints, contains a distinct number of elements.
Original entry on oeis.org
1, 1, 2, 1, 3, 4, 2, 5, 6, 3, 7, 4, 8, 2, 9, 5, 10, 11, 6, 12, 3, 13, 14, 7, 15, 4, 16, 17, 8, 18, 2, 19, 20, 21, 9, 22, 5, 23, 24, 10, 25, 11, 26, 6, 27, 28, 12, 29, 30, 13, 31, 14, 32, 7, 33, 15, 34, 35, 36, 16, 37, 17, 38, 8, 39, 18, 40, 41, 19, 42, 43, 20
Offset: 1
a(1)=1; no pair of terms exists yet.
a(2)=1 creates the pair [1, 1], which encloses 0 elements. This means that no consecutive equal values can occur again, since this would create another set of 0 elements.
a(3)=2 because if a(3)=1, this would create a second pair enclosing 0 elements.
a(4)=1 creates two new sets: [1, 2, 1], enclosing 1 element {2}, and [1, 1, 2, 1], enclosing 2 elements {1, 2}.
a(5) cannot be 1 as this would again create a pair enclosing 0 elements [1,1]. 2 would create the pair [2, 1, 2] which encloses 1 element {1}, which has been impossible since a(4). So a(5)=3, which has not occurred before.
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See Links section.
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from itertools import islice
def agen(): # generator of terms
e, a = set(), []
while True:
an, allnew = 0, False
while not allnew:
allnew, an, ndset = True, an+1, set()
for i in range(len(a)):
if an == a[i]:
nd = len(set(a[i+1:]))
if nd in e or nd in ndset: allnew = False; break
ndset.add(nd)
yield an; a.append(an); e |= ndset
print(list(islice(agen(), 72))) # Michael S. Branicky, Oct 25 2023
A363654
Lexicographically earliest sequence of positive integers such that the n-th pair of identical terms encloses exactly a(n) terms.
Original entry on oeis.org
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
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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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
A363708
Lexicographically earliest sequence of positive integers such that the n-th pair of consecutive equal values are separated by a(n) terms, with pairs numbered according to the position of the first term in the pair.
Original entry on oeis.org
1, 2, 1, 3, 2, 3, 4, 5, 2, 4, 6, 5, 7, 2, 5, 4, 8, 7, 9, 10, 2, 4, 10, 11, 8, 12, 2, 10, 13, 14, 15, 13, 11, 12, 14, 16, 2, 17, 10, 2, 18, 15, 12, 13, 19, 20, 21, 14, 17, 22, 23, 17, 10, 24, 2, 13, 12, 15, 20, 12, 25, 21, 17, 26, 14, 27, 28, 22, 29, 30, 31, 32
Offset: 1
The first pair (1,2,1) encloses 1 term because a(1)=1.
The second pair (2,1,3,2) encloses 2 terms because a(2)=2.
The third pair (3,2,3) encloses 1 term because a(3)=1.
The fourth pair (2,3,4,5,2) encloses 3 terms because a(4)=3.
In constructing the sequence, we must consider whether a number forms a pair with some earlier term, and if so, whether this leads to a contradiction. If every existing term leads to a contradiction, then the smallest number not yet in the sequence is used, as in a(7)=4.
a(7)=4 because if a(7)=1, we get (1,2,1,3,2,3,1). This would have the third pair in the sequence enclose 3 terms even though a(3)=1, which is satisfied by the pair (3,2,3).
If a(7)=2: (1,2,1,3,2,3,2) would have the fourth pair (2,3,2) enclose 1 term, which is impossible because a(4)=3 means the fourth pair encloses 3 terms.
If a(7)=3: (1,2,1,3,2,3,3) would have the pair (3,3) enclosing 0 terms, yet 0 is never a term.
a(7)=4 without contradiction since 4 has not yet appeared in the sequence.
From _Kevin Ryde_, Jun 27 2023: (Start)
Backtracking can be illustrated at a(20) != 8. The following candidates are the actual a(1..19) followed by prospective 8 for a(20),
n = ... 14 15 16 17 18 19 20
a(n) = ... 2, 5, 4, 8, 7, 9, 8 <-- attempt a(20)=8
This is consistent if all of 2,5,4 pair with later terms so the 8's are pair number 14, gap a(14)=2.
But it turns out each possible value at a(21) makes such pair number 14 impossible, so no later terms can let this start work, and so must go back and change something.
(Logic could show in advance that 2,5 cannot both pair, but other impossible combinations might be more complex.)
(End)
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
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.
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
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.
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