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

1,2

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.

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.

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 *)

Prog

(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

Crossrefs

Cf. A084383, A033627, A026474.

Sequence in context: A180482 A193460 A114801 * A055933 A188650 A132578

Adjacent sequences: A114799 A114800 A114801 * A114803 A114804 A114805

Keyword

base,nonn

Author

Jonathan Vos Post, Feb 18 2006

Extensions

Corrected and edited by Giovanni Resta, Jun 15 2016

Status

approved