A396837 Lexicographically earliest sequence of distinct nonnegative integers whose binary values, when written on a square spiral, one bit per square, high order bits first, contain no pair of 0 bits that are a knight's move apart.

0, 1, 2, 3, 5, 7, 6, 15, 14, 12, 4, 29, 8, 9, 13, 10, 31, 11, 59, 30, 62, 63, 126, 21, 119, 127, 17, 22, 19, 25, 26, 23, 27, 16, 43, 238, 125, 124, 53, 55, 118, 255, 61, 254, 111, 123, 250, 42, 38, 51, 253, 24, 46, 236, 20, 28, 506, 47, 93, 45, 52, 40, 239

Offset

0,3

Comments

Although the terms equal to powers of 2 are delayed relative to similar size terms, e.g. a(2214) = 128, it is conjectured that the sequence contains all nonnegative integers.

Links

Scott R. Shannon, Table of n, a(n) for n = 0..5000

Scott R. Shannon, Image of the first 5000 terms on the square spiral. Zoom in to see the details. The 1 bits are black while the 0 bits are light gray. The bits that comprise each term are connected by a red line.

Example

The spiral begins:
.
                                .
    0---1---1---0---1---1---1   1
    |                       |   |
    0   1---1---0---1---1   0   1
    |   |               |   |   |
    0   1   1---0---1   1   0   1
    |   |   |       |   |   |   |
    1   1   1   0---1   1   1   1
    |   |   |           |   |   |
    0   1   1---0---1---1   0   1
    |   |                   |   |
    0   1---1---0---1---1---0   0
    |                           |
    1---1---1---0---1---1---0---1
.
a(0) = 0 by definition.
a(2) = 2 = 10_2 as the 4th square, which contains the 0 bit of 10_2, is not a knight's move away from the 1st square, the only other square containing a 0 bit.
a(4) = 5 = 101_2 as the 8th square, which contains the central 0 bit of 101_2, is not a knight's move away from the 1st or 4th squares, the other squares containing a 0 bit. Note that a(4) cannot equal 4 = 100_2 as that would place a 0 bit on the 9th square which is a knight's move away from the 4th square.

Crossrefs

Cf. A396836 (start at 1), A007088, A396603, A396604, A396607, A396608.

Sequence in context: A353955 A318954 A296375 * A381013 A306923 A366856

Adjacent sequences: A396829 A396833 A396836 * A396850 A396851 A396855

Keyword

nonn,base,new

Author

Scott R. Shannon, Jun 07 2026

Status

approved