A352713 Lexicographically earliest sequence of distinct nonnegative integers such that the binary expansions of two consecutive terms have no common 1, and the least value not yet in the sequence appears as soon as possible.

0, 1, 2, 4, 3, 8, 5, 16, 6, 24, 7, 32, 9, 20, 10, 36, 11, 48, 12, 18, 13, 64, 14, 80, 15, 96, 17, 40, 19, 72, 21, 104, 22, 128, 23, 160, 25, 68, 26, 100, 27, 192, 28, 34, 29, 224, 30, 256, 31, 320, 33, 76, 35, 88, 37, 136, 38, 144, 39, 208, 41, 84, 42, 132, 43

Offset: 0

Author

Rémy Sigrist, Mar 30 2022

Keywords

• nonn
• base

Comments

To build the sequence:

- we start with a(0) = 0, and then iteratively:

- let v be the last value, and u be the least value not yet in the sequence,

- if v AND u = 0, then the next value is u (AND denotes the bitwise AND operator),

- otherwise the next values are w and then u where w is chosen as small as possible.

This sequence is a variant of A109812 where we repeatedly force the least unseen value to appear as soon as possible.

By design, this is a permutation of the nonnegative integers (with inverse A352714).

Examples

			The first terms are (stars correspond to "w" terms):
  n   a(n)  bin(a(n))  w
  --  ----  ---------  -
   0     0          0
   1     1          1
   2     2         10
   3     4        100  *
   4     3         11
   5     8       1000  *
   6     5        101
   7    16      10000  *
   8     6        110
   9    24      11000  *
  10     7        111
  11    32     100000  *
  12     9       1001
  13    20      10100  *
  14    10       1010
  15    36     100100  *
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, Colored logarithmic scatterplot of the first 2^16 terms (the color denotes the parity of n: blue for even, red for odd)
• Rémy Sigrist, PARI program
• Index entries for sequences related to binary expansion of n
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A109812, A352714 (inverse).

Programs

PARI

See Links section.

Python

from math import gcd
from itertools import count, islice
def agen(): # generator of terms
    aset, v, u = {0}, 0, 1; yield 0
    for n in count(1):
        if v&u != 0:
            w = u + 1
            while w in aset or v&w != 0 or w&u != 0: w += 1
            aset.add(w); yield w
        v = u; aset.add(v); yield v
        while u in aset: u += 1
print(list(islice(agen(), 65))) # Michael S. Branicky, Jun 24 2022