A036496 Number of lines that intersect the first n points on a spiral on a triangular lattice. The spiral starts at (0,0), goes to (1,0) and (1/2, sqrt(3)/2) and continues counterclockwise.

0, 3, 5, 6, 7, 8, 9, 9, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 27, 27, 28, 28, 28, 28, 29, 29, 29, 29, 29, 30

Offset

0,2

Comments

The triangular lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called a hexagonal lattice.

Conjecture: a(n) is half the minimal perimeter of a polyhex of n hexagons. - Winston C. Yang (winston(AT)cs.wisc.edu), Apr 06 2002. This conjecture follows from the Brunvoll et al. reference. - Sascha Kurz, Mar 17 2008

From a spiral of n triangular lattice points, we can get a polyhex of n hexagons with min perimeter by replacing each point on the spiral by a hexagon. - Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002

References

J. Bornhoft, G. Brinkmann, J. Greinus, Pentagon-hexagon-patches with short boundaries, European J. Combin. 24 (2003), 517-529.

F. Harary and H. Harborth, Extremal animals, Journal of Combinatorics, Information, & System Sciences, Vol. 1, 1-8, (1976).

W. C. Yang, Maximal and minimal polyhexes, manuscript, 2002.

W. C. Yang, PhD thesis, Computer Sciences Department, University of Wisconsin-Madison, 2003.

J. Brunvoll, B.N. Cyvin and S.J Cyvin, More about extremal animals, Journal of Mathematical Chemistry Vol. 12 (1993), pp. 109-119

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Formula

If n >= 1, a(n) = ceiling(sqrt(12n - 3)). - Winston C. Yang (winston(AT)cs.wisc.edu), Apr 06 2002

Example

For n=3 the 3 points are (0,0), (1,0), (1/2, sqrt(3)/2) and there are 3 lines: 2 horizontal, 2 sloping at 60 degrees and 2 at 120 degrees, so a(3)=6.

Mathematica

Join[{0}, Ceiling[Sqrt[12*Range[80]-3]]] (* Harvey P. Dale, May 26 2017 *)

Crossrefs

Cf. A001399, A038147.

Sequence in context: A004220 A202308 A079058 * A196112 A009004 A005527

Adjacent sequences: A036493 A036494 A036495 * A036497 A036498 A036499

Keyword

nonn,easy,nice

Author

Mario VELUCCHI (mathchess(AT)velucchi.it)

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Sep 29 2000

Status

approved