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.

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, -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, 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

Offset: 1

Author

Attila Kiss, Aug 11 2023

Keywords

• sign
• base
• look

Comments

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.

Examples

			Sequence A: 0, 1, 3, 4, 5, 14, 15, 16, 17, 18, 20, 21, 22, ...
Sequence B: 1, 2, 3, 4, 8, 10, 13, 15, 16, 22, 23, 25, 30, ...
    {a(n)}: 1, 1, 0, 0, 3, -4, -2, -1, -1,  4,  3,  4,  8, ...
		

Links

• Attila Kiss, Table of n, a(n) for n = 1..17483.
• Attila Kiss, Picture of the first ~17000 terms.
• Attila Kiss, Java program to generate terms.

Crossrefs

Cf. A365017, A365018.

Programs

Mathematica

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 *)

Python

from itertools import islice
def g(s=0): # helper generator for sequences A (s=0) and B (s=1)
    cn, bins, concats = s, {bin(s)[2:]}, set()
    while True:
        yield cn
        while (bn:=bin(cn:=cn+1)[2:]) in concats: pass
        concats |= {bn+bi for bi in bins} | {bi+bn for bi in bins}
        bins.add(bn)
def agen(): # generator of terms
    A, B = g(s=0), g(s=1)
    yield from (Bn - An for An, Bn in zip(A, B))
print(list(islice(agen(), 78))) # Michael S. Branicky, Nov 01 2023

Formula

a(n) = A365018(n) - A365017(n).