A114801 2-concatenation-free sequence starting (1,2).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 121, 123, 124, 125, 126, 127, 128, 129, 131, 132, 134, 135, 136, 137, 138, 139, 141, 142, 143, 145, 146, 147, 148, 149, 151, 152, 153, 154, 156, 157, 158

Offset: 1

Author

Jonathan Vos Post, Feb 18 2006

Keywords

• base
• easy
• nonn

Comments

Starting with the terms (1,2) this sequence consists of minimum increasing terms such that no term is the concatenation of any two previous distinct terms. The next consecutive number skipped after 121 is 122 = Concatenate(1, 22). This is the analog of a 2-Stöhr sequence with concatenation (base 10) substituting for addition. A033627 "0-additive sequence: not the sum of any previous pair" is another name for the 2-Stöhr sequence.

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Stöhr Sequence.
• Rémy Sigrist, PARI program for A114801

Crossrefs

Cf. A084383, A033627.

Programs

Mathematica

conc[x_, y_] := FromDigits@ Flatten@ IntegerDigits[{x, y}]; L = {1, 2}; cc = {12, 21}; Do[k = 1 + Max@L; While[MemberQ[cc, k], k++]; cc = Union[cc, conc[#, k] & /@ L, conc[k, #] & /@ L]; AppendTo[L, k];, {65}]; L (* Giovanni Resta, Jun 15 2016 *)

PARI

See Links section.

Python

from itertools import count, islice
def agen(): # generator of terms
    cats1, cats2, an, s = {"1", "2"}, {"12", "21"}, 3, "3"
    yield from [1, 2]
    while True:
        yield an
        cats2 |= {s + c for c in cats1} | {c + s for c in cats1}
        cats1.add(s)
        while (s:=str(an)) in cats1 or s in cats2:
            an += 1
print(list(islice(agen(), 59))) # Michael S. Branicky, Feb 01 2024

Formula

a(0) = 1, a(1) = 2, for n>2: a(n) = least k > a(n-1) such that k is not an element of {Concatenate(a(i), a(j))} for any distinct a(i) <= a(n-1) and a(j) <= a(n-1).

Extensions

Data corrected by Giovanni Resta, Jun 14 2016