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

Author

Neal Gersh Tolunsky, Feb 13 2024

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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

Extensions

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