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A364871 a(n) = B(n) - A(n), where A(1) = 0, B(1) = 1, and sequences A and B are the lexicographically earliest sequences that start with their respective first terms and contain no term whose binary expansion is the concatenation of any two earlier terms in that sequence.

%I A364871 #33 Nov 01 2023 22:00:53
%S A364871 1,1,0,0,3,-4,-2,-1,-1,4,3,4,8,8,11,11,11,1,2,-2,-1,-11,-11,-12,-12,
%T A364871 -7,-8,-5,-1,-2,-2,2,-2,0,0,1,3,6,9,9,14,10,12,13,15,19,21,22,22,25,
%U A364871 25,20,17,11,13,8,10,6,6,1,0,4,4,4,9,8,8,1,1,1,1,-6,-12,-11,-15,-21,-18,-20
%N A364871 a(n) = B(n) - A(n), where A(1) = 0, B(1) = 1, and sequences A and B are the lexicographically earliest sequences that start with their respective first terms and contain no term whose binary expansion is the concatenation of any two earlier terms in that sequence.
%C A364871 Empirical observation: the graph looks like hills in a repeating fractal-like pattern; each peak is about 1.4 times as high as the previous one.
%H A364871 Attila Kiss, <a href="/A364871/b364871.txt">Table of n, a(n) for n = 1..17483</a>.
%H A364871 Attila Kiss, <a href="/A364871/a364871.jpg">Picture of the first ~17000 terms</a>.
%H A364871 Attila Kiss, <a href="/A364871/a364871.java.txt">Java program to generate terms</a>.
%F A364871 a(n) = A365018(n) - A365017(n).
%e A364871 Sequence A: 0, 1, 3, 4, 5, 14, 15, 16, 17, 18, 20, 21, 22, ...
%e A364871 Sequence B: 1, 2, 3, 4, 8, 10, 13, 15, 16, 22, 23, 25, 30, ...
%e A364871     {a(n)}: 1, 1, 0, 0, 3, -4, -2, -1, -1,  4,  3,  4,  8, ...
%t A364871 conc[x_, y_] := FromDigits[Flatten@IntegerDigits[{x, y}, 2], 2]; f[n_, m_] := f[n, m] = If[n == 1, m, Module[{k = f[n - 1, m] + 1, v = Array[f[#, m] &, n - 1], c}, c = conc @@@ Select[Tuples[v, {2}], UnsameQ @@ # &]; While[! FreeQ[c, k], k++]; k]]; a[n_] := f[n, 1] - f[n, 0]; Array[a, 80] (* _Amiram Eldar_, Sep 29 2023 *)
%o A364871 (Python)
%o A364871 from itertools import islice
%o A364871 def g(s=0): # helper generator for sequences A (s=0) and B (s=1)
%o A364871     cn, bins, concats = s, {bin(s)[2:]}, set()
%o A364871     while True:
%o A364871         yield cn
%o A364871         while (bn:=bin(cn:=cn+1)[2:]) in concats: pass
%o A364871         concats |= {bn+bi for bi in bins} | {bi+bn for bi in bins}
%o A364871         bins.add(bn)
%o A364871 def agen(): # generator of terms
%o A364871     A, B = g(s=0), g(s=1)
%o A364871     yield from (Bn - An for An, Bn in zip(A, B))
%o A364871 print(list(islice(agen(), 78))) # _Michael S. Branicky_, Nov 01 2023
%Y A364871 Cf. A365017, A365018.
%K A364871 sign,base,look
%O A364871 1,5
%A A364871 _Attila Kiss_, Aug 11 2023