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A320495
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Coordination sequence of thinnest 5-neighbor packing of the plane with congruent hexagons with respect to a point of type A.
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7
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1, 4, 6, 12, 14, 19, 22, 28, 32, 34, 39, 44, 46, 52, 54, 58, 62, 69, 69, 75, 77, 82, 87, 93, 92, 98, 100, 107, 111, 117, 114, 122, 123, 132, 134, 140, 137, 146, 148, 156, 157, 163, 160, 171, 172, 180, 179, 187, 183, 196, 195, 203, 202, 211, 208, 220, 218, 226
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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"5-neighbor" means that each hexagon has a point in common with exactly five other hexagons.
This packing is actually the thinnest 5-neighbor packing in the plane using any centrally symmetric congruent polygons.
More formally, this sequence is the coordination sequence of the vertex-edge graph of the packing with respect to a vertex of type A. (The automorphism group of the tiling has four orbits on vertices, indicated by the letters A, B, C, D in the figure.)
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REFERENCES
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William Moser and Janos Pach, Research Problems in Discrete Geometry: Packing and Covering, DIMACS Technical Report 93-32, May 1993. See Fig. 19.1b, page 32. There is an error in the figure: the hexagon at the right of the bottom row should not be shaded. The figure shown here is correct.
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LINKS
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N. J. A. Sloane, The packing and its graph. (The hexagons are shaded, the base point is marked A, and the green dots indicate the centers of large empty hexagrams.)
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FORMULA
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Based on the b-file, this appears to have g.f. = f/g, where
f = -2*x^24+x^23-x^22-x^21+5*x^20+6*x^19+4*x^18+5*x^17+
13*x^16+16*x^15+9*x^14+8*x^13+13*x^12+16*x^11+11*x^10
+9*x^9+14*x^8+13*x^7+12*x^6+11*x^5+9*x^4+8*x^3+5*x^2+4*x+1
and
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PROG
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(PARI) See Links section.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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