cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A375207 a(1)=1; thereafter a(n) is the smallest k for which the subsequence a(n-k..n-1) has a distinct sequence of first differences from that of any other subsequence of the sequence thus far.

Original entry on oeis.org

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

Views

Author

Neal Gersh Tolunsky, Oct 16 2024

Keywords

Comments

In other words, a(n) is the length of the shortest subsequence ending at a(n-1) which has unique first differences among all first differences of subsequences of the sequence thus far.

Examples

			a(8)=2 because the length-2 subsequence a(6..7) = 4,3 has the shortest unique first differences (-1) of any other subsequence in the sequence thus far.
a(9)=3 because the length-3 subsequence a(6..8) = 4,3,2 likewise has the shortest unique first differences (-1,-1). No shorter subsequence ending in a(8) with unique first differences exists in the sequence thus far. We cannot have, for example, a(7..8) = 3,2 since we saw a subsequence with the same first differences in the previous example, where a(6..7) = 4,3 has first difference (-1).

Links

Crossrefs

Cf. A377079, A376937.

Programs

  • Python
    from itertools import count, islice
def fd(t): return tuple(t[i]-t[i-1] for i in range(1, len(t)))
def agen(): # generator of terms
    a, R, maxL = [1], set(), 0  # maxL = max length of first diff.'s stored
    for n in count(1):
        yield a[-1]
        for k in range(1, n+1):
            if k > maxL:  # must increase length of first diff.'s stored
                maxL += 1
                R.update(fd(a[i:i+maxL]) for i in range(n-maxL))
            if fd(a[-k:]) not in R:
                break
        an = k
        R.update(fd(a[-i:]) for i in range(1, maxL+1))
        a.append(an)
print(list(islice(agen(), 90))) # Michael S. Branicky, Oct 16 2024

A377079 a(1)=1; thereafter a(n) is the smallest k for which the subsequence a(n-k..n-1) has a distinct ordinal transform from that of any other subsequence of the sequence thus far.

Original entry on oeis.org

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

Views

Author

Neal Gersh Tolunsky, Oct 15 2024

Keywords

Comments

In other words, a(n) is the length of the shortest subsequence ending at a(n-1) which has a unique ordinal transform among all ordinal transforms of subsequences of the sequence thus far. Alternatively, this is (the length of the longest subsequence ending at a(n-1) whose ordinal transform has occurred before as that of another subsequence) plus 1.

Examples

			a(8)=4 because the length-4 subsequence a(4..7) = 2,3,4,3 has the shortest unique ordinal transform (1,1,1,2), which does not occur elsewhere as the ordinal transform of any other subsequence in the sequence thus far. No shorter subsequence ending in a(7) with a unique ordinal transform exists in the sequence thus far. For example, a(8) cannot be 3 because the length-3 subsequence a(5..7) = 3,4,3 has the same ordinal transform (1,1,2) as that of the subsequence a(2..4) = 1,2,2.

Links

Crossrefs

Cf. A376937, A375207.

Programs

  • Python
    from itertools import count, islice
def ot(t): # after ORDINAL at https://oeis.org/wiki/Ordinal_transform
    c={}; return tuple(c.setdefault(x, c.pop(x, 0)+1) for x in t)
def agen(): # generator of terms
    a, R, maxL = [1], set(), 0  # maxL = max length of ot's stored
    for n in count(1):
        yield a[-1]
        for k in range(1, n+1):
            if k > maxL:  # must increase length of ot's stored
                maxL += 1
                R.update(ot(a[i:i+maxL]) for i in range(n-maxL))
            if ot(a[-k:]) not in R:
                break
        an = k
        R.update(ot(a[-i:]) for i in range(1, maxL+1))
        a.append(an)
print(list(islice(agen(), 90))) # Michael S. Branicky, Oct 15 2024

A378876 a(1)=1; thereafter a(n) is the smallest k for which the subsequence a(n-k..n-1) has a distinct multiset from that of any other subsequence of the sequence thus far.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 3, 1, 3, 3, 2, 2, 2, 3, 6, 1, 2, 3, 3, 6, 3, 4, 2, 2, 3, 5, 1, 2, 3, 5, 5, 2, 2, 3, 6, 5, 2, 3, 5, 5, 5, 3, 5, 5, 6, 3, 5, 9, 1, 2, 3, 4, 4, 2, 3, 5, 4, 2, 3, 5, 5, 6, 6, 2, 2, 3, 5, 8, 1, 2, 3, 4, 5, 5, 3, 5, 5, 6, 7, 1, 2, 3, 4, 5, 6, 3, 5, 5
Offset: 1

Views

Author

Neal Gersh Tolunsky, Jan 17 2025

Keywords

Comments

In other words, a(n) is the length of the shortest subsequence ending at a(n-1) which has a unique multiset among all multisets of subsequences of the sequence thus far. Alternatively, this is (the length of the longest subsequence ending at a(n-1) whose multiset has occurred before as that of another subsequence) plus 1.

Examples

			a(15) = 6 because the length-6 subsequence a(9..14) =  3,3,2,2,2,3 has the shortest unique multiset, which does not occur elsewhere as the multiset of any other subsequence in the sequence thus far. No shorter subsequence ending in a(14) with a unique ordinal transform exists in the sequence thus far. For example, a(15) cannot be 5 because the length-5 subsequence a(10..14) = 3,2,2,2,3 has the same multiset as that of the subsequence a(9..13) = 3,3,2,2,2.

Crossrefs

Cf. A376937, A375207, A377079.
Showing 1-3 of 3 results.

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