A370263 Lexicographically earliest sequence such that each subsequence enclosed by a pair of equal values, excluding the endpoints, has a unique sum.

1, 1, 2, 1, 2, 3, 1, 2, 4, 1, 5, 3, 6, 1, 2, 4, 7, 2, 3, 5, 6, 4, 5, 7, 8, 1, 3, 5, 2, 6, 4, 9, 10, 7, 1, 3, 8, 11, 4, 2, 8, 9, 12, 1, 3, 6, 4, 7, 11, 10, 2, 5, 8, 13, 14, 2, 9, 15, 7, 1, 3, 4, 11, 13, 12, 16, 1, 5, 14, 6, 8, 10, 9, 4, 17, 2, 10, 3, 18, 11, 16

Offset

1,3

Comments

Two consecutive equal values enclose no terms, which have a sum of 0, and thus after [a(1), a(2)] = [1, 1] no consecutive equal values will occur again.

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

Links

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

Example

a(2)=1 creates the pair [a(1), a(2)] = [1, 1], which gives the unique sum of 0.
a(4)=1 creates two unique sums: [1,2,1] -> [2] = sum of 2 and [1,1,2,1] -> [1,2] = sum of 3.
a(8)=2 creates two unique sums: [2,3,1,2] -> [3,1] = sum of 4 and [2,1,2,3,1,2] -> [1,2,3,1] = sum of 7.

Prog

(Python)
from itertools import islice
def agen(): # generator of terms
    s, a = set(), []
    while True:
        an, allnew = 0, False
        while not allnew:
            allnew, an, sn = True, an+1, set()
            for i in range(len(a)):
                if an == a[i]:
                    t = sum(a[i+1:])
                    if t in s or t in sn: allnew = False; break
                    sn.add(t)
        yield an; a.append(an); s |= sn
print(list(islice(agen(), 81))) # Michael S. Branicky, Feb 14 2024

Crossrefs

Cf. A370264 (including endpoints), A366624, A366493, A366631, A366625.

Sequence in context: A334430 A214614 A265692 * A376308 A380911 A194976

Adjacent sequences: A370260 A370261 A370262 * A370264 A370265 A370266

Keyword

nonn

Author

Neal Gersh Tolunsky, Feb 13 2024

Extensions

a(16) and beyond from Michael S. Branicky, Feb 14 2024

Status

approved