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.

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

Author

Neal Gersh Tolunsky, Oct 16 2024

Keywords

• nonn

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

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
• Neal Gersh Tolunsky, Ordinal Transform of 500000 terms

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