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%I #17 Jun 10 2019 00:10:59
%S 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,
%T 18,18,18,19,19,19,20,20,20,21,21,21,21,22,22,22,23,23,23,23,24,24,24,
%U 24,25,25,25,25,26,26,26,26,27,27,27,27,27,28,28,28,28,29,29,29,29,29,30
%N 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.
%C The triangular lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called a hexagonal lattice.
%C 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
%C 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
%D J. Bornhoft, G. Brinkmann, J. Greinus, Pentagon-hexagon-patches with short boundaries, European J. Combin. 24 (2003), 517-529.
%D F. Harary and H. Harborth, Extremal animals, Journal of Combinatorics, Information, & System Sciences, Vol. 1, 1-8, (1976).
%D W. C. Yang, Maximal and minimal polyhexes, manuscript, 2002.
%D W. C. Yang, PhD thesis, Computer Sciences Department, University of Wisconsin-Madison, 2003.
%D J. Brunvoll, B.N. Cyvin and S.J Cyvin, More about extremal animals, Journal of Mathematical Chemistry Vol. 12 (1993), pp. 109-119
%H Harvey P. Dale, <a href="/A036496/b036496.txt">Table of n, a(n) for n = 0..1000</a>
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%F If n >= 1, a(n) = ceiling(sqrt(12n - 3)). - Winston C. Yang (winston(AT)cs.wisc.edu), Apr 06 2002
%e 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.
%t Join[{0},Ceiling[Sqrt[12*Range[80]-3]]] (* _Harvey P. Dale_, May 26 2017 *)
%Y Cf. A001399, A038147.
%K nonn,easy,nice
%O 0,2
%A Mario VELUCCHI (mathchess(AT)velucchi.it)
%E More terms from Larry Reeves (larryr(AT)acm.org), Sep 29 2000
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