A365018 -id:A365018 - cp's OEIS frontend

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

Attila Kiss, Aug 11 2023

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.

			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, ...

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.

Cf. A365017, A365018.

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 *)

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

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

Original entry on oeis.org

0, 1, 3, 4, 5, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 38, 39, 46, 47, 60, 61, 64, 65, 66, 68, 69, 70, 72, 73, 74, 80, 81, 82, 84, 85, 86, 88, 89, 90, 96, 97, 98, 100, 101, 104, 105, 106, 108, 109, 115, 119, 126, 127, 134, 135, 142, 143, 151, 156, 157, 158, 166, 167, 174

Offset: 1

Attila Kiss, Aug 16 2023

a(1)=0 is taken to be a single 0 bit when concatenating.

			The number 2 is excluded because its binary expansion is "10", which is the concatenation of a(1)="1" and a(0)="0".
The number 19 is excluded because its binary expansion is "10011", which is the concatenation of a(4)="100" and a(3)="11".

Attila Kiss, Java code to generate terms.

Cf. A365018, A364871.

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 *)

from itertools import islice
def agen(): # generator of terms
    an, bins, concats = 0, {"0"}, set()
    while True:
        yield an
        while (bn:=bin(an:=an+1)[2:]) in concats: pass
        concats |= {bn+bi for bi in bins} | {bi+bn for bi in bins}
        bins.add(bn)
print(list(islice(agen(),62))) # Michael S. Branicky, Sep 29 2023

cp's OEIS Frontend

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.

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