A114802 3-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, 131, 141, 151, 161, 171, 181, 191, 200, 212, 232, 242, 252, 262, 272, 282, 292, 300, 313, 323, 343, 353, 363, 373, 383, 393, 400, 414, 424, 434, 454

Offset: 1

Author

Jonathan Vos Post, Feb 18 2006

Keywords

• base
• nonn

Comments

Starting with the terms (1,2) this sequence consists of minimum increasing integer terms such that no term is the concatenation of any two or three previous distinct terms. The next consecutive numbers skipped after 121 are 122 = Concatenate(1,22) and 123 = Concatenate(1,2,3). This is the analog of a 3-Stöhr sequence with concatenation (base 10) substituting for addition. A026474 is a 3-Stöhr sequence.

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Stöhr Sequence.

Crossrefs

Cf. A084383, A033627, A026474.

Programs

Mathematica

conc[w_] := Flatten[ (FromDigits /@ Flatten /@ IntegerDigits /@ (Permutations[#]) &) /@ Subsets[w, {2, 3}]]; up = 10^3; L = {1, 2, 3}; cc = conc[L]; Do[k = 1 + Max@L; While[MemberQ[cc, k], k++]; If[k > up, Break[]]; Do[cc = Union[cc, Select[ conc[{k, L[[i]], L[[j]]}], # <= up &]], {i, Length[L]}, {j, i - 1}]; AppendTo[L, k], {60}]; L (* Giovanni Resta, Jun 15 2016 *)

Python

from itertools import islice
def incats(s, L, k):
    if s == "": return True
    if k == 0: return False
    return any(s.startswith(w) and incats(s[len(w):], L[:i]+L[i+1:], k-1) for i, w in enumerate(L))
def agen(): # generator of terms
    L, an, s = ["1", "2"], 3, "3"
    yield from [1, 2]
    while True:
        yield an
        L.append(s)
        while incats((s:=str(an)), L, 3):
            an += 1
print(list(islice(agen(), 70))) # 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(h),a(i),a(j)]} or {Concatenate[a(i),a(j)]} for any three distinct a(h), a(i), and a(j), where h, i, j < n.

Extensions

Corrected and edited by Giovanni Resta, Jun 15 2016