A274820 Spiral constructed on the nodes of the infinite triangular net in which each term is the least nonnegative integer such that no diagonal contains a repeated term.

0, 1, 2, 1, 2, 1, 2, 0, 3, 0, 4, 3, 5, 3, 4, 5, 3, 4, 6, 5, 6, 7, 4, 6, 5, 7, 6, 3, 0, 6, 5, 7, 0, 6, 7, 5, 4, 8, 1, 3, 6, 8, 1, 9, 7, 8, 2, 4, 9, 8, 2, 10, 11, 8, 9, 10, 12, 3, 8, 9, 7, 10, 9, 2, 4, 8, 5, 10, 2, 11, 9, 11, 0, 10, 7, 8, 6, 0, 9, 7, 10, 12, 7, 1, 4, 8, 5, 11, 1, 10, 12, 9, 5, 11, 10, 13, 12, 11, 13, 14

Offset: 0

Author

Omar E. Pol, Jul 09 2016

Keywords

• nonn
• look

Comments

Also spiral constructed on the infinite hexagonal grid in which each term is the least nonnegative integer such that no diagonal of successive adjacent cells contains a repeated term. Every number is located in the center of a hexagonal cell. Every cell is also the center of three diagonals of successive adjacent cells.

Presumably every line of cells with slope a multiple of 60 degrees (not necessarily passing through the central cell) is a permutation of the nonnegative numbers. See A296343-A296348 for the spokes through the central cell. - N. J. A. Sloane, Dec 12 2017

Examples

			Illustration of initial terms as a spiral:
.
.                   9 - 4 - 2 - 8 - 7
.                  /                 \
.                 8   3 - 6 - 7 - 5   9
.                /   /             \   \
.               2   0   5 - 3 - 4   6   1
.              /   /   /         \   \   \
.            10   6   3   1 - 2   0   4   8
.            /   /   /   /     \   \   \   \
.          11   5   4   2   0 - 1   3   7   6
.            \   \   \   \         /   /   /
.             8   7   5   1 - 2 - 0   6   3
.              \   \   \             /   /
.               9   0   3 - 4 - 6 - 5   1
.                \   \                 /
.                10   6 - 7 - 5 - 4 - 8
.                  \
.                  12 - 3 - 8 - 9 - 7
.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..120400
• F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
• Rémy Sigrist, PARI program for A274820
• Rémy Sigrist, Colored illustration of the first 200 windings of the spiral (where the color is a function of a(n))
• N. J. A. Sloane, Illustration of initial terms drawn as a spiral on the hexagonal grid (the starting cell is marked in black).

Crossrefs

Cf. A001477, A269526, A274528 (square array), A274641 (spiral on the square grid), A274650 (right triangle), A274821, A274920, A274921, A275606, A275610, A296339.

A296342 says when n first appears.

See A296343-A296348 for the spokes.

Programs

PARI

See Links section.

Formula

a(n) = A274821(n) - 1.