A396603 Lexicographically earliest sequence of distinct nonegative integers whose binary values, when written on a square spiral, one bit per square, high order bits first, contain no pair of 1 bits that are a knight's move apart.

0, 1, 32, 2, 64, 3, 1024, 16, 8, 68, 4, 6, 5, 48, 10, 128, 12, 18, 4096, 34, 40, 256, 512, 130, 66, 42, 17, 9, 96, 33, 24, 14, 11, 258, 136, 520, 260, 8192, 2048, 72, 138, 131072, 100, 528, 2080, 20, 514, 160, 36, 65, 104, 73, 19, 67, 21, 26, 168, 74, 162, 1028, 288

Offset

0,3

Comments

After a(0) = 0 when determining the next term in the sequence one must not only ensure that it contains no 1 bits that are a knight's move apart from any previous 1 bit, including the earlier 1 bits in the term itself, but also that the next square after the end of the term is not a knight's move apart from any previous 1 bit as the next term must begin with a 1 bit.

It is conjectured that the sequence contains all nonnegative integers.

Links

Scott R. Shannon, Table of n, a(n) for n = 0..1000

Scott R. Shannon, Image of the first 1000 terms on the square spiral. 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---0---0---0---0---1   1
    |                       |   |
    0   0---0---0---0---0   0   1
    |   |               |   |   |
    0   1   0---0---1   0   0   0
    |   |   |       |   |   |   |
    1   1   0   0---1   1   0   1
    |   |   |           |   |   |
    0   1   0---0---1---0   0   0
    |   |                   |   |
    0   0---0---0---0---0---0   1
    |                           |
    0---1---0---0---1---0---0---1
.
a(0) = 0 by definition.
a(1) = 1 as there are no 1 other bits on the spiral, and the third square, the next square after the end of a(1) = 1, can contain a 1 bit as it is not a knight's move away from any existing 1 bit.
a(2) = 32. The 5th square cannot contain a 1 bit as it is a knight's move away from the 2nd square which contains a 1 bit, so a(2) cannot be 2 or 3. Likewise the 6th, 7th and 8th square cannot contain a 1 bit, leaving 32 as the smallest number that contains no 1 bits in these squares while allowing a 1 bit in the 9th square to start the fourth term.

Crossrefs

Cf. A396604 (start at 1), A007088, A396607, A396608, A394166.

Sequence in context: A037932 A285417 A303328 * A298921 A070631 A070623

Adjacent sequences: A396600 A396601 A396602 * A396604 A396605 A396606

Keyword

nonn,base

Author

Scott R. Shannon, May 30 2026

Status

approved