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Search: id:A036496
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| A036496 |
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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. |
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+0
2
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| 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; listen)
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OFFSET
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0,2
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COMMENT
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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 (sascha.kurz(AT)uni-bayreuth.de), 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
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REFERENCES
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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
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LINKS
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G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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FORMULA
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If n >= 1, a(n) = ceil(sqrt(12n - 3)). - Winston C. Yang (winston(AT)cs.wisc.edu), Apr 06 2002
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EXAMPLE
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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 degs and 2 at 120 degs, so a(3)=6.
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CROSSREFS
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Cf. A001399, A038147.
Sequence in context: A070083 A004220 A079058 this_sequence A009004 A005527 A009005
Adjacent sequences: A036493 A036494 A036495 this_sequence A036497 A036498 A036499
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Mario VELUCCHI (mathchess(AT)velucchi.it)
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Sep 29 2000
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