A370708 a(1)=1; thereafter a(n) is the smallest number > a(n-1) such that no two triples of earlier terms in arithmetic progression have the same common difference.

1, 2, 3, 5, 6, 8, 12, 13, 15, 16, 21, 23, 28, 32, 37, 38, 40, 45, 47, 61, 63, 70, 73, 80, 81, 91, 96, 100, 103, 105, 116, 123, 128, 134, 138, 150, 156, 157, 175, 179, 181, 190, 207, 210, 214, 217, 226, 240, 243, 252, 256, 265, 275, 281, 283, 289, 292, 293, 308, 315

Offset

1,2

Comments

A triple consists of three distinct values in a(1), a(2), ..., a(n).

By definition, no arithmetic progression of length > 3 can occur in the sequence.

What is the density of this sequence?

Links

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

Example

4 is not a term in the sequence because it would create the arithmetic progression (2,3,4), which has the same common difference (1) as the previously occurring triple (1,2,3).
9 is not a term because it would create the arithmetic progression (3,6,9), which has the same common difference (3) as the previously occurring (2,5,8).

Prog

(Python)
from itertools import islice
def cd(k, alst, dset, diff_dict):
    newdset = set()
    for a in alst:
        if k-a in diff_dict[a]:
            if k-a in dset:
                return False
            else:
                newdset.add(k-a)
    return True, newdset
def agen(): # generator of terms
    alst, dset, an = [1, 2, 3], {1}, 3
    yield from alst
    diff_dict = {1: set(), 2: {1}, 3: {1, 2}}
    while True:
        k = an+1
        while not (ans:=cd(k, alst, dset, diff_dict)): k += 1
        dset.update(ans[1])
        an = k
        diff_dict[k] = {an-a for a in alst}
        alst.append(an)
        yield an
print(list(islice(agen(), 60))) # Michael S. Branicky, Mar 30 2024

Crossrefs

Cf. A003278.

Sequence in context: A120768 A271109 A293033 * A002384 A096176 A330748

Adjacent sequences: A370705 A370706 A370707 * A370709 A370710 A370711

Keyword

nonn

Author

Neal Gersh Tolunsky, Mar 25 2024

Extensions

a(15) and beyond from Michael S. Branicky, Mar 30 2024

Status

approved