A114802 - OEIS

%I #19 Feb 02 2024 15:18:29

%S 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,

%T 99,100,121,131,141,151,161,171,181,191,200,212,232,242,252,262,272,

%U 282,292,300,313,323,343,353,363,373,383,393,400,414,424,434,454

%N 3-concatenation-free sequence starting (1,2).

%C 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.

%H Michael S. Branicky, <a href="/A114802/b114802.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StoehrSequence.html">Stöhr Sequence.</a>

%F 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.

%t 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 *)

%o (Python)

%o from itertools import islice

%o def incats(s, L, k):

%o     if s == "": return True

%o     if k == 0: return False

%o     return any(s.startswith(w) and incats(s[len(w):], L[:i]+L[i+1:], k-1) for i, w in enumerate(L))

%o def agen(): # generator of terms

%o     L, an, s = ["1", "2"], 3, "3"

%o     yield from [1, 2]

%o     while True:

%o         yield an

%o         L.append(s)

%o         while incats((s:=str(an)), L, 3):

%o             an += 1

%o print(list(islice(agen(), 70))) # _Michael S. Branicky_, Feb 01 2024

%Y Cf. A084383, A033627, A026474.

%K base,nonn

%O 1,2

%A _Jonathan Vos Post_, Feb 18 2006

%E Corrected and edited by _Giovanni Resta_, Jun 15 2016