A397519 Lexicographically earliest sequence of distinct nonnegative integers whose binary plot has no two 0 bits a knight's move apart.

0, 1, 2, 3, 6, 4, 7, 15, 5, 8, 13, 31, 21, 10, 23, 11, 29, 14, 37, 30, 61, 26, 53, 27, 55, 42, 63, 43, 85, 46, 87, 47, 12, 28, 79, 95, 9, 25, 127, 111, 20, 62, 119, 18, 183, 126, 22, 38, 54, 190, 86, 110, 94, 174, 118, 175, 214, 191, 215, 58, 93, 59, 117, 106

Offset

0,3

Comments

A variation of A394166 where the 0 bits, as opposed to the 1 bits, cannot be a knight's move apart.

It is conjectured that all nonnegative numbers appear.

Links

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Example

The terms, with their binary values, begin:
  n   a(n)   A007088(a(n))
  --  ----  --------------
   0     0               0
   1     1               1
   2     2              10
   3     3              11
   4     6             110
   5     4             100
   6     7             111
   7    15            1111
   8     5             101
   9     8            1000
  10    13            1101
  11    31           11111
  12    21           10101
  13    10            1010
  14    23           10111
  15    11            1011
  16    29           11101
  17    14            1110
  18    37          100101
  19    30           11110
  20    61          111101
.

Prog

(Python)
from itertools import count, islice
def comp(k): return (~k)&((1<<k.bit_length())-1)
def agen(): # generator of terms
    aset, m, an2, an1 = {0, 1}, 2, 0, 1
    yield from [0, 1]
    for n in count(2):
        can1, can2 = comp(an1), comp(an2)
        mask = (can1<<2)|(can1>>2)|(can2<<1)|(can2>>1)
        an2, an1 = an1, next(k for k in count(m) if not (comp(k)&mask or k in aset))
        yield an1
        aset.add(an1)
        while m in aset: m += 1
print(list(islice(agen(), 64))) # Michael S. Branicky, Jun 30 2026

Crossrefs

Cf. A394166, A007088, A396836, A109812, A396694.

Sequence in context: A273338 A099900 A193903 * A138728 A291604 A082332

Adjacent sequences: A397515 A397516 A397518 * A397520 A397521 A397522

Keyword

nonn,base

Author

Scott R. Shannon, Jun 29 2026

Status

approved