A370264 Lexicographically earliest sequence such that each subsequence enclosed by a pair of equal values, including the endpoints, has a unique sum.

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

Offset

1,3

Comments

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 2.
a(4)=1 creates two unique sums: [1,2,1] = sum of 4 and [1,1,2,1] = sum of 5.
a(8)=3 creates one unique sum: [3,2,1,3] = sum of 9.

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:]) + 2*an
                    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 (excluding endpoints), A366493, A366624, A366631, A366625.

Sequence in context: A115624 A076291 A275015 * A211189 A194968 A194980

Adjacent sequences: A370261 A370262 A370263 * A370265 A370266 A370267

Keyword

nonn

Author

Neal Gersh Tolunsky, Feb 13 2024

Extensions

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

Status

approved