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A365017 a(n) is the least nonnegative integer not already in the sequence whose binary expansion is not the concatenation of any two earlier terms.

%I A365017 #46 Nov 05 2023 15:00:13
%S A365017 0,1,3,4,5,14,15,16,17,18,20,21,22,24,25,26,27,38,39,46,47,60,61,64,
%T A365017 65,66,68,69,70,72,73,74,80,81,82,84,85,86,88,89,90,96,97,98,100,101,
%U A365017 104,105,106,108,109,115,119,126,127,134,135,142,143,151,156,157,158,166,167,174
%N A365017 a(n) is the least nonnegative integer not already in the sequence whose binary expansion is not the concatenation of any two earlier terms.
%C A365017 a(1)=0 is taken to be a single 0 bit when concatenating.
%H A365017 Attila Kiss, <a href="/A365017/a365017.java.txt">Java code to generate terms</a>.
%e A365017 The number 2 is excluded because its binary expansion is "10", which is the concatenation of a(1)="1" and a(0)="0".
%e A365017 The number 19 is excluded because its binary expansion is "10011", which is the concatenation of a(4)="100" and a(3)="11".
%t A365017 conc[x_, y_] := FromDigits[Flatten@IntegerDigits[{x, y}, 2], 2]; a[1] = 0; a[n_] := a[n] = Module[{k = a[n - 1] + 1, v = Array[a, n - 1], c}, c = conc @@@ Select[Tuples[v, {2}], UnsameQ @@ # &]; While[! FreeQ[c, k], k++]; k]; Array[a, 60] (* _Amiram Eldar_, Sep 29 2023 *)
%o A365017 (Python)
%o A365017 from itertools import islice
%o A365017 def agen(): # generator of terms
%o A365017     an, bins, concats = 0, {"0"}, set()
%o A365017     while True:
%o A365017         yield an
%o A365017         while (bn:=bin(an:=an+1)[2:]) in concats: pass
%o A365017         concats |= {bn+bi for bi in bins} | {bi+bn for bi in bins}
%o A365017         bins.add(bn)
%o A365017 print(list(islice(agen(),62))) # _Michael S. Branicky_, Sep 29 2023
%Y A365017 Cf. A365018, A364871.
%K A365017 nonn,base
%O A365017 1,3
%A A365017 _Attila Kiss_, Aug 16 2023