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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 21 14:54 EDT 2020. Contains 337272 sequences. (Running on oeis4.)