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
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