A397519 - OEIS

A397519

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

%I #12 Jun 30 2026 10:46:37

%S 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,

%T 63,43,85,46,87,47,12,28,79,95,9,25,127,111,20,62,119,18,183,126,22,

%U 38,54,190,86,110,94,174,118,175,214,191,215,58,93,59,117,106

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

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

%C It is conjectured that all nonnegative numbers appear.

%H Scott R. Shannon, <a href="/A397519/b397519.txt">Table of n, a(n) for n = 0..10000</a>

%e The terms, with their binary values, begin:

%e n a(n) A007088(a(n))

%e -- ---- --------------

%e 0 0 0

%e 1 1 1

%e 2 2 10

%e 3 3 11

%e 4 6 110

%e 5 4 100

%e 6 7 111

%e 7 15 1111

%e 8 5 101

%e 9 8 1000

%e 10 13 1101

%e 11 31 11111

%e 12 21 10101

%e 13 10 1010

%e 14 23 10111

%e 15 11 1011

%e 16 29 11101

%e 17 14 1110

%e 18 37 100101

%e 19 30 11110

%e 20 61 111101

%e .

%o (Python)

%o from itertools import count, islice

%o def comp(k): return (~k)&((1<<k.bit_length())-1)

%o def agen(): # generator of terms

%o aset, m, an2, an1 = {0, 1}, 2, 0, 1

%o yield from [0, 1]

%o for n in count(2):

%o can1, can2 = comp(an1), comp(an2)

%o mask = (can1<<2)|(can1>>2)|(can2<<1)|(can2>>1)

%o an2, an1 = an1, next(k for k in count(m) if not (comp(k)&mask or k in aset))

%o yield an1

%o aset.add(an1)

%o while m in aset: m += 1

%o print(list(islice(agen(), 64))) # _Michael S. Branicky_, Jun 30 2026

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

%K nonn,base

%O 0,3

%A _Scott R. Shannon_, Jun 29 2026