A392194 Lexicographically earliest sequence of positive numbers on a square spiral such that each pair of orthogonal nearest neighbor numbers is distinct.

1, 1, 2, 3, 3, 4, 2, 5, 6, 2, 7, 8, 1, 9, 5, 7, 4, 6, 8, 10, 1, 11, 4, 9, 12, 1, 13, 6, 11, 14, 2, 15, 3, 8, 9, 10, 3, 12, 6, 14, 5, 16, 2, 17, 3, 13, 9, 11, 18, 2, 19, 5, 15, 7, 7, 20, 1, 21, 4, 14, 12, 15, 10, 16, 4, 18, 5, 17, 9, 20, 6, 22, 1, 23, 4, 19, 7, 16

Offset

1,3

Comments

For the terms studied three of the diagonals leading away from the central square are composed of alternating values of 1 and 2, while one of the north-west diagonals is composed of alternating values of 3 and 4.

After 5 million terms the smallest pair including 1 that has not occurred is 1:6706, implying it is likely all numbers paired with 1 eventually appear. However in this range the pair 2:2 has not occurred, while the next lowest pair including 2 that has not occurred is 2:6706. This implies that 2:2 may never occur, although this is unknown. Likewise 4:4 and 6:10 have not occurred.

Occasionally a number occurs that is smaller than the average of its surrounding neighbors which causes subsequent neighbors in the outer rings of numbers to also change. This leads to linear diagonal areas of lesser and great values to appear randomly in the resulting pattern of numbers - see the attached images.

Links

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Scott R. Shannon, Image of the first 10000 terms on square the spiral. The number colors are graduated across the spectrum, from red to violet, to show their relative size. Zoom in to see the numbers.

Scott R. Shannon, Image of the first 1000000 terms on the square spiral. Image may need to be downloaded for best viewing.

Example

The square spiral begins:
                       .
                       .
   4---7---5---9---1  14
   |               |   |
   6   3---3---2   8  11
   |   |       |   |   |
   8   4   1---1   7   6
   |   |           |   |
  10   2---5---6---2  13
   |                   |
   1--11---4---9--12---1
.
a(3) = 2 as a(3) has 1 as an orthogonal neighbor while a(1) and a(2) have already formed the othogonal pair 1-1, so a(3) cannot be 1.
a(4) = 3 as a(4) has 1 and 2 as orthogonal neighbors while orthogonal pairs 1-1 and 1-2 have already occurred, so a(4) cannot be 1 or 2.
a(5) = 3 as a(5) has 3 as an orthogonal neighbor while orthogonal pairs 1-3 and 2-3 have already occurred, so a(5) cannot be 1 or 2.
a(6) = 4 as a(6) has 1 and 3 as orthogonal neighbors while orthogonal pairs 1-1, 1-2, 1-3, 3-3 have already occurred, so a(6) cannot be 1, 2 or 3.
a(7) = 2 as a(7) has 4 as an orthogonal neighbor while orthogonal pair 1-4 has already occurred, so a(7) cannot be 1.

Crossrefs

Cf. A274640, A307834, A355314, A355270, A355271.

Sequence in context: A269526 A106448 A295280 * A307838 A159909 A176004

Adjacent sequences: A392191 A392192 A392193 * A392195 A392196 A392197

Keyword

nonn

Author

Scott R. Shannon, Mar 05 2026

Status

approved