A330896 -id:A330896 - cp's OEIS frontend

A330897 a(n) is the least k such that A330896(k) = n.

1, 3, 8, 12, 21, 35, 43, 71, 85, 107, 133, 138, 173, 220, 240, 290, 320, 355, 405, 430, 512, 559, 593, 622, 727, 731, 846, 901, 940, 1071, 1107, 1207, 1279, 1389, 1490, 1544, 1657, 1699, 1768, 1870, 1935, 2049, 2198, 2346, 2468, 2478, 2612, 2693, 2817, 2919

Offset: 1

Rémy Sigrist, May 01 2020

nonn

			The first terms of A330896 are:
           n|  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 ...
  A330896(n)|  1  1  2  1  2  2  1  3  2  3  1  4  2  3  3  1  4  2  4  3  5 ...
- hence a(1) = 1, a(2) = 3, a(3) = 8, a(4) = 12, a(5) = 21.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A330897

Cf. A330896.

C
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See Links section.
```

A330896(a(n)) = n.

A366493 Lexicographically earliest sequence such that each subsequence enclosed by two equal terms is distinct.

Original entry on oeis.org

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

Offset: 1

Neal Gersh Tolunsky, Oct 10 2023

A new value is always followed by 1.

			a(2)=1 because the subsequence (1,1) has not occurred before.
a(8)=3 because every smaller number would form a subsequence that has occurred already. a(8) cannot be 1 because this would make the subsequence (1,1), which was seen before at i=1,2. a(8) cannot be 2 because then we would have the subsequence (2,1,2) for a second time (first at i=3-5): 1,1,2,1,2,2,1,2

Samuel Harkness, Table of n, a(n) for n = 1..10000
Samuel Harkness, MATLAB program

Cf. A330896.

MATLAB
```
See Links section.
```

from itertools import islice
def agen(): # generator of terms
    m, a = set(), []
    while True:
        an, allnew = 0, False
        while not allnew:
            allnew, an, mn = True, an+1, set()
            for i in range(len(a)):
                if an == a[i]:
                    t = tuple(a[i:]+[an])
                    if t in m or t in mn: allnew = False; break
                    mn.add(t)
        yield an; a.append(an); m |= mn
print(list(islice(agen(), 111))) # Michael S. Branicky, Dec 06 2024

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

Neal Gersh Tolunsky, Oct 17 2023

nonn

The word 'set' means that every element is unique. For example, the set {1,1,2} contains 2 elements (not 3).
Note that we are considering sets between every pair of equal values, not just those that appear consecutively.
Two consecutive values enclose 0 terms, and thus after [a(1), a(2)] = [1, 1], no consecutive equal values occur again.

			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.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program

Cf. A337226 (with nondistinct terms counted), A330896, A363757, A366631.

PARI
```
See Links section.
```

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

More terms from Rémy Sigrist, Oct 25 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.

A366624 Lexicographically earliest sequence of positive integers such that each subsequence enclosed by two equal terms, not including the endpoints, is distinct.

Original entry on oeis.org

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

Offset: 1

Samuel Harkness, Oct 14 2023

nonn

Every positive integer occurs infinitely many times in the sequence.
The subsequence between any two equal terms is unique. For example, consecutive values "A B A" prevents "C B C" because the subsequence "B" would be repeated between equal terms.
Two consecutive values create the empty subsequence, for this reason after {a(1), a(2)} = {1, 1}, no consecutive values will ever occur again.
A new value is always followed by 1.

			For a(9), we first try 1. If a(9) were 1, {a(8)} = {2} would be bounded by a(7) = a(9) = 1, but {2} is already bounded by a(2) = a(4) = 1.
Next, try 2. Note this would create consecutive values at {a(8), a(9)}, enclosing the empty subsequence, but this already occurred at {a(1), a(2)}.
Next, try 3. If a(9) were 3, {a(7), a(8)} = {1, 2} would be bounded by a(6) = a(9) = 3, but {1, 2} is already bounded by a(1) = a(4) = 1.
Next, try 4. 4 has yet to appear, so a(9) = 4.

Michael S. Branicky, Table of n, a(n) for n = 1..6804 (terms 1..1000 from Neal Gersh Tolunsky)
Samuel Harkness, MATLAB program.
Kevin Ryde, C Code.

Cf. A366625 (with distinct multisets), A366631 (with distinct sets), A366493 (including endpoints), A330896, A366691.

C
```
/* See links */
```
MATLAB
```
See Links section.
```

from itertools import islice
def agen(): # generator of terms
    m, a = set(), []
    while True:
        an, allnew = 0, False
        while not allnew:
            allnew, an, mn = True, an+1, set()
            for i in range(len(a)):
                if an == a[i]:
                    t = tuple(a[i+1:])
                    if t in m or t in mn: allnew = False; break
                    mn.add(t)
        yield an; a.append(an); m |= mn
print(list(islice(agen(), 87))) # Michael S. Branicky, Jan 15 2024

A366625 Lexicographically earliest sequence of positive integers such that each multiset enclosed by two equal terms, excluding the endpoints, is distinct.

Original entry on oeis.org

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

Offset: 1

Samuel Harkness, Oct 14 2023

nonn

Every positive integer occurs infinitely many times in the sequence.
The multiset between any two equal terms is unique. For example: once consecutive values "A B C A" occur, both "D B C D" and "D C B D" can never occur, because the multiset "B C" would be repeated between equal terms.
Two consecutive values enclose the empty multiset. For this reason, after [a(1), a(2)] = [1, 1], no consecutive equal values will occur again.
A new value is always followed by 1.
The sequence first differs from A366624 at a(15).

			a(15) = 5: a(15) cannot be 1 since this would form the empty multiset with a(14) = 1. a(15) cannot be 2 because this would form the multiset [2 1 2] = {1}, which already occurred at [2 1 2] = {1}. a(15) cannot be 3 because this would form the multiset [3 2 1 3] = {1, 2}, which already occurred at [1 1 2 1] = {1, 2}. a(15) cannot be 4 because this would form the multiset [4 1 2 3 2 1 4] = {1, 1, 2, 2, 3}, which already occurred at [1 1 2 1 2 3 1] = {1, 1, 2, 2, 3}. a(15) = 5 because 5 is a first occurrence and thus creates no new multisets.
For this sequence, the multisets between k and all other occurrences of k must be checked. The first instance such that this is the sole reason for restricting a possible value is when considering 2 for a(27).
a(27) != 2 since 2 there would cause two enclosed multisets with the same 5 terms (in different order, which doesn't matter for a multiset),
  n    =   15      19    22      26 27
  a(n) =  1 5 1 2 3 4 1 2 3 4 2 1 5 [2]
            |-------|     |-------|
There are also instances where overlapping conflicting regions are the sole reason for restricting a possible value.
a(74) != 5 since 5 there would cause two enclosed multisets with the same 20 terms,
  n    =  38                              54
  a(n) = 2 6 1 2 3 4 5 1 2 4 3 7 1 2 3 4 5 3 1
           |----------------------------------
                                           |--
  n    =  57                              73
  a(n) = 2 6 2 1 3 4 5 7 1 2 3 4 5 6 1 2 3 4[5]
         --|
         ----------------------------------|

Michael S. Branicky, Table of n, a(n) for n = 1..7522
Samuel Harkness, MATLAB program.
Neal Gersh Tolunsky, Ordinal Transform of 5000 terms.

Cf. A366624, A366631, A366493, A330896.

MATLAB
```
See Links section.
```

from itertools import islice
def agen(): # generator of terms
    m, a = set(), []
    while True:
        an, allnew = 0, False
        while not allnew:
            allnew, an, mn = True, an+1, set()
            for i in range(len(a)):
                if an == a[i]:
                    t = tuple(sorted(a[i+1:]))
                    if t in m or t in mn: allnew = False; break
                    mn.add(t)
        yield an; a.append(an); m |= mn
print(list(islice(agen(), 87))) # Michael S. Branicky, Oct 25 2023

A370577 Lexicographically earliest sequence such that for any value m, the number of distinct values between a pair of consecutive m's is distinct.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 2, 3, 3, 4, 1, 2, 3, 4, 3, 4, 4, 5, 1, 2, 3, 4, 5, 5, 6, 1, 2, 3, 4, 5, 6, 4, 5, 6, 5, 6, 6, 7, 1, 2, 3, 4, 5, 6, 7, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 5, 6, 7, 8, 7, 8, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 7, 8, 9, 9, 10, 1, 2, 3, 4, 5, 6, 7

Offset: 1

Neal Gersh Tolunsky, Feb 22 2024

nonn

			The first terms with the number of distinct values enclosed by m = 1..4 below:
   n|  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
...
a(n)|  1  1  2  1  2  2  3  1  2  3  3  4  1  2  3  4  3  4  4  5  1  2  3
...
----+---------------------------------------------------------------------
 1's|     0,    1,          2,             3,                      4,
...
 2's|              1, 0,       2,             3,                      4,
...
 3's|                             2, 0,          3,    1,                4,
...
 4's|                                               3,    1, 0,
...

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf.A330896, A366691, A281511.

from itertools import islice
def agen(): # generator of terms
    e, a = set(), []
    while True:
        an, allnew = 0, False
        while not allnew:
            allnew, an, nd = True, an+1, None
            for i in range(len(a)-1, -1, -1):
                if an == a[i]:
                    nd = len(set(a[i+1:]))
                    if (an, nd) in e: allnew = False
                    break
        yield an; a.append(an); e.add((an, nd))
print(list(islice(agen(), 86))) # Michael S. Branicky, Feb 22 2024

More terms from Michael S. Branicky, Feb 22 2024

A379381 a(1)=1, a(2)=2; thereafter, a(n) is the smallest positive integer such that for any value k, the number of distinct values between a pair of k's is distinct, counting k itself.

Original entry on oeis.org

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

Offset: 1

Neal Gersh Tolunsky, Dec 21 2024

nonn

Note that we are considering every pair of equal values, not just those that appear consecutively.

			a(7)=4: We cannot have a(7)=1 here because this would make a(1..7) = 1, 2, 1, 2, 3, 3, 1 enclose the same number of terms as a(3..7) = 1, 2, 3, 3, 1 (3 distinct values). We cannot have a(7)=2 because this would mean a(4..7) = 2, 3, 3, 2 encloses 2 values, which we had at a(2..4) = 2, 1, 2. a(7) cannot be 3 because this would repeat a(5-6) = 3, 3 with a(6-7) = 3, 3, again enclosing 1 distinct value. So a(7) = 4 without restriction.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Cf. A330896, A366691, A370577, A281511.

cp's OEIS Frontend

A330897 a(n) is the least k such that A330896(k) = n.

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A366493 Lexicographically earliest sequence such that each subsequence enclosed by two equal terms is distinct.

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A366691 Lexicographically earliest sequence such that each set of terms enclosed by two equal values, excluding the endpoints, contains a distinct number of elements.

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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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A366624 Lexicographically earliest sequence of positive integers such that each subsequence enclosed by two equal terms, not including the endpoints, is distinct.

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A366625 Lexicographically earliest sequence of positive integers such that each multiset enclosed by two equal terms, excluding the endpoints, is distinct.

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MATLAB

Python

A370577 Lexicographically earliest sequence such that for any value m, the number of distinct values between a pair of consecutive m's is distinct.

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Python

Extensions

A379381 a(1)=1, a(2)=2; thereafter, a(n) is the smallest positive integer such that for any value k, the number of distinct values between a pair of k's is distinct, counting k itself.

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