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

Author

Neal Gersh Tolunsky, Mar 25 2024

Keywords

• nonn

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?

Examples

			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).
		

Links

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

Crossrefs

Cf. A003278.

Programs

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

Extensions

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