A364871 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

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.

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.

FORMULA

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

EXAMPLE

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, ...

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 *)

PROG

(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

CROSSREFS

Cf. A365017, A365018.

Sequence in context: A200125 A091528 A236679 * A096392 A332964 A105825

Adjacent sequences: A364868 A364869 A364870 * A364872 A364873 A364874

KEYWORD

sign,base,look

AUTHOR

Attila Kiss, Aug 11 2023

STATUS

approved