A328818 -id:A328818 - cp's OEIS frontend

A307012 Second coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. First and third coordinates are given in A307011 and A345978.

0, 0, 1, 1, 0, -1, -1, -1, 0, 1, 2, 2, 2, 1, 0, -1, -2, -2, -2, -2, -1, 0, 1, 2, 3, 3, 3, 3, 2, 1, 0, -1, -2, -3, -3, -3, -3, -3, -2, -1, 0, 1, 2, 3, 4, 4, 4, 4, 4, 3, 2, 1, 0, -1, -2, -3, -4, -4, -4, -4, -4, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 4

Offset: 0

Hugo Pfoertner, Mar 19 2019

The coordinate system can be described using 3 axes that pass through spiral point 0 and one of points 1, 2 or 3. Along each axis, one of the coordinates is 0. a(n) is the signed distance from spiral point n to the axis that passes through point 1. The distance is measured along either of the lines through point n that are parallel to one of the other 2 axes and the sign is such that point 2 has positive distance. - Peter Munn, Jul 13 2021

We can use this coordinate with the first coordinate to form an oblique coordinate system, in which each coordinate maps to an oblique coordinate vector parallel to the axis along which the other coordinate is 0. See the figure with nonperpendicular axes in the Barile link. When both of these coordinates are positive, the oblique coordinate vectors make a 60-degree angle with each other. [Made more specific by Peter Munn, Jul 19 2021]

Hugo Pfoertner, Table of n, a(n) for n = 0..10150
Margherita Barile, Oblique Coordinates, entry in Eric Weisstein's World of Mathematics.
HandWiki, Hexagonal Lattice.
Peter Munn, Illustration of signed distance of spiral points.
Wikipedia, Signed distance function.
Index entries for sequences related to coordinates of 2D curves

Cf. A307011, A307013, A328818, A334493, A345435, A345978.

Name revised by Peter Munn, Jul 08 2021

A307013 Third coordinate (asymmetric variant) in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. First and second coordinates are given in A307011 and A307012.

Original entry on oeis.org

0, 1, 1, 0, -1, -1, 0, 1, 2, 2, 2, 1, 0, -1, -2, -2, -2, -1, 0, 1, 2, 3, 3, 3, 3, 2, 1, 0, -1, -2, -3, -3, -3, -3, -2, -1, 0, 1, 2, 3, 4, 4, 4, 4, 4, 3, 2, 1, 0, -1, -2, -3, -4, -4, -4, -4, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 4, 3, 2, 1, 0, -1, -2

Offset: 0

Hugo Pfoertner, Mar 19 2019

From Peter Munn, Jul 11 2021: (Start)
The points of the spiral are equally the points of a hexagonal lattice, the points of an isometric (triangular) grid and the center points of the cells of a honeycomb (regular hexagonal tiling or grid). The coordinate system can be described using three "0-axes" that pass through spiral point 0 and one of points 1, 2 or 3. These 0-axes are the lines along which one of the coordinates is 0.
a(n), the 3rd coordinate, is the signed distance from spiral point n to the coordinate's 0-axis, which passes through points 0 and 3. The distance is measured along either of the lines through point n that are parallel to one of the other 2 axes and the sign is such that point 1 has positive distance. This 3rd coordinate is the sum of the other 2. In the symmetric variant of the coordinate system, the 3rd coordinate has the opposite sense, so that the 3 coordinates sum to 0. See A345978.
We can use any 2 of the 3 coordinates to form an oblique coordinate system, in which each of the 2 coordinates specifies vectors parallel to the other coordinate's 0-axis. This means the direction of the oblique coordinate vectors depends on the choice of the other coordinate - see the illustration of coordinate pairing in the links. When both coordinates are positive, an oblique coordinate vector derived from this sequence makes a 120-degree angle with the vector derived from the other sequence; however, when A307011 and A307012 are used together, the angle is 60 degrees.
Pairing with A307012 can be viewed as follows. Let omega = -1/2 + i*sqrt(3)/2, a primitive cube root of unity. Then f(n) = a(n) + omega*A307012(n) embeds the spiral in the complex plane with spiral points 0 and 1 embedded at 0 and 1 (so that the points of the spiral embed as the Eisenstein integers, as used for A345435).
(End)

Hugo Pfoertner, Table of n, a(n) for n = 0..10092
Margherita Barile, Oblique Coordinates, entry in Eric Weisstein's World of Mathematics.
HandWiki, Hexagonal Lattice.
Peter Munn, Illustration of coordinate pairing.
Eric Weisstein's World of Mathematics, Eisenstein Integer
Wikipedia, Signed distance function.

Cf. A307011, A307012, A328818, A345435, A345978.

A334492 is effectively this "3rd coordinate" for a different sequence of points on a hexagonal lattice.

a(n) = A307011(n) + A307012(n). - Peter Munn, Jul 04 2021

A307011 First coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. Second and third coordinates are given in A307012 and A345978.

Original entry on oeis.org

0, 1, 0, -1, -1, 0, 1, 2, 2, 1, 0, -1, -2, -2, -2, -1, 0, 1, 2, 3, 3, 3, 2, 1, 0, -1, -2, -3, -3, -3, -3, -2, -1, 0, 1, 2, 3, 4, 4, 4, 4, 3, 2, 1, 0, -1, -2, -3, -4, -4, -4, -4, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5

Offset: 0

Hugo Pfoertner, Mar 19 2019

From Peter Munn, Jul 22 2021: (Start)
The points of the spiral are equally the points of a hexagonal lattice, the points of an isometric (triangular) grid and the center points of the cells of a honeycomb (regular hexagonal tiling or grid). The coordinate system can be described using 3 axes that pass through spiral point 0 and one of points 1, 2 or 3. Along each axis, one of the coordinates is 0.
a(n) is the signed distance from spiral point n to the axis that passes through point 2. The distance is measured along either of the lines through point n that are parallel to one of the other 2 axes and the sign is such that point 1 has positive distance.
This coordinate can be paired with either of the other coordinates to form oblique coordinates as described in A307012. Alternatively, all 3 coordinates can be used together, symmetrically, as described in A345978.
There is a negated variant of the 3rd coordinate, which is the conventional sense of this coordinate for specifying (with the 2nd coordinate) the Eisenstein integers that can be the points of the spiral when it is embedded in the complex plane. See A307013.
(End)

Hugo Pfoertner, Table of n, a(n) for n = 0..10034
Margherita Barile, Oblique Coordinates, entry in Eric Weisstein's World of Mathematics.
HandWiki, Hexagonal Lattice.
Peter Munn, Illustration of signed distance of spiral points.
Hugo Pfoertner, Illustration of A307012 vs A307011, spiral.
Hugo Pfoertner, Illustration of A345978 vs A307011, spiral.
Wikipedia, Signed distance function.

Cf. A307012, A307013, A307014, A307016, A307017, A328818, A345978.
Numbers on the spokes of the spiral: A000567, A028896, A033428, A045944, A049450, A049451.
Positions on the spiral that correspond to Eisenstein primes: A345435.

r=-1;d=-1;print1(m=0,", ");for(k=0,8,for(j=1,r,print1(s,", "));if(k%2,,m++;r++);for(j=-m,m+1,if(d*j>=-m,print1(s=d*j,", ")));d=-d)

Name revised by Peter Munn, Jul 08 2021

A345764 Number the tiles of a regular hexagonal tiling from 0 in a spiral. Consider perpendicular axes, X and Y, through the center of tile 0, one of which passes through the center of tile 1. Define a set of equivalence classes of tiles with respect to reflections about X and Y. a(n) is the smallest number of a tile in the same equivalence class as tile n.

Original entry on oeis.org

0, 1, 2, 2, 1, 2, 2, 7, 8, 7, 10, 11, 10, 7, 8, 7, 10, 11, 10, 19, 20, 21, 20, 19, 24, 25, 25, 24, 19, 20, 21, 20, 19, 24, 25, 25, 24, 37, 38, 39, 40, 39, 38, 37, 44, 45, 46, 45, 44, 37, 38, 39, 40, 39, 38, 37, 44, 45, 46, 45, 44, 61, 62, 63, 64, 65, 64, 63, 62, 61

Offset: 0

Peter Munn, Jun 26 2021

nonn

The sense of the spiral (clockwise/counterclockwise) and its orientation are not significant, but for the purpose of illustration, we depict a counterclockwise spiral with its first step towards the right side of the page.
Equivalence classes contain a maximum of 4 tiles. This happens when tile m's reflection about axis X is a different tile, m_x, and these 2 tiles' reflections about axis Y are 2 further tiles, m_y and m_xy, to give an equivalence class {m, m_x, m_y, m_xy}. Some equivalence classes are smaller, because a tile is its own reflection about an axis, X or Y, that passes through the center of the tile.
The Wichmann reference describes bijections from certain unique factorization domains to the hexagonal tiling. Align the spiral with the mapping so that domain identities 0 and 1 map to tiles 0 and 1 respectively. If two integers from one of the domains map to tiles in the same equivalence class, then they share the same status as units, primes or composites.

			Illustration of the relative positions of tiles on the spiral, marking the n-th tile on the spiral by a(n) to denote its equivalence class:
.
.              24 -- 25 -- 25 -- 24
.              /                   \
.             /                     \
.           19    10 -- 11 -- 10    19
.           /     /             \     \
.          /     /               \     \
.        20     7     2 --- 2     7    20
.        /     /     /       \     \     \
.       /     /     /         \     \     \
.     21     8     1     0 --- 1     8    21
.       \     \     \               /     /
.        \     \     \             /     /
.        20     7     2 --- 2 --- 7    20
.          \     \                     /
.           \     \                   /
.           19    10 -- 11 -- 10 -- 19
.             \
.              \
.              24 -- 25 -- 25 -- 24
.
Recall that the underlying tile numbers count steps along the spiral from 0. When we follow the spiral in the illustration above and encounter a number m, which denotes an equivalence class, for the first time, this is also at tile number m.
Tile 1 maps to itself (as does tile 4) when reflected about the axis through the centers of tiles 0 and 1 (horizontal as shown above). Tiles 1 and 4 map to each other when reflected about the perpendicular (vertical) axis. So tiles 1 and 4 form an equivalence class, and the smallest number of a tile in this class is 1. So a(1) = 1 and a(4) = 1.

Eric Weisstein's World of Mathematics, Hexagonal Grid.
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019.

Cf. A274820, A307012, A307013, A328818.

a(n) = min({m : |A307012(m)| = |A307012(n)| and |A328818(m)| = |A328818(n)|}).

a(n) = min({m : |A307012(n)| = |A307012(m)| and |2*A307013(n) - A307012(n)| = |2*A307013(m) - A307012(m)|}).

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A307012 Second coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. First and third coordinates are given in A307011 and A345978.

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A307013 Third coordinate (asymmetric variant) in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. First and second coordinates are given in A307011 and A307012.

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A307011 First coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. Second and third coordinates are given in A307012 and A345978.

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