A114801 - OEIS

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

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OFFSET

1,2

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

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).

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 *)

PROG

(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: