A396607 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 diagonally adjacent.

0, 1, 2, 3, 64, 128, 4, 6, 5, 8, 9, 7, 256, 80, 10, 1024, 11, 12, 16, 17, 32, 20, 13, 14, 512, 18, 272, 34, 21, 2048, 15, 36, 22, 40, 19, 42, 320, 23, 24, 1280, 144, 130, 258, 136, 25, 640, 65, 26, 48, 66, 38, 33, 28, 138, 1026, 35, 41, 160, 132, 72, 514, 192, 74

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 diagonally adjacent to 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 diagonally adjacent to 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---0---1---0---0---0---1   0
    |                       |   |
    1   0---0---0---1---0   1   0
    |   |               |   |   |
    1   0   1---0---1   0   0   0
    |   |   |       |   |   |   |
    1   0   1   0---1   0   1   1
    |   |   |           |   |   |
    1   0   1---0---0---0   0   0
    |   |                   |   |
    1   0---1---0---0---1---1   1
    |                           |
    0---0---0---0---0---0---0---0
.
a(0) = 0 by definition.
a(1) = 1 as there are no 1 other bits on the spiral, and the 3rd square, the next square after the end of a(1) = 1, can contain a 1 bit as it is only diagonally adjacent to a(0) = 0.
a(4) = 64. This term must be 4 or more but it cannot end on the 9th, 10th, 11th, or 12th square as the following square cannot be 1 due to the 2nd and 3rd square containing 1 bits. The first term that is 4 or more and ends on the 13th square, allowing the 14th square to contain a 1 bit, is 64.

Crossrefs

Cf. A396608 (start at 1), A007088, A396603, A396604, A394166.

Sequence in context: A280582 A037426 A004854 * A356797 A356785 A015169

Adjacent sequences: A396604 A396605 A396606 * A396608 A396609 A396610

Keyword

nonn,base

Author

Scott R. Shannon, May 31 2026

Status

approved