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

0,3

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).

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

Example

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  *

Prog

(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

Crossrefs

Cf. A109812, A352714 (inverse).

Sequence in context: A120242 A322864 A355436 * A054427 A363537 A232563

Adjacent sequences: A352710 A352711 A352712 * A352714 A352715 A352716

Keyword

nonn,base

Author

Rémy Sigrist, Mar 30 2022

Status

approved