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A320496
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Coordination sequence of thinnest 5-neighbor packing of the plane with congruent hexagons with respect to a point of type B.
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7
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1, 2, 6, 9, 16, 17, 22, 25, 31, 34, 37, 43, 47, 49, 56, 56, 60, 65, 72, 74, 79, 79, 83, 90, 97, 97, 102, 101, 108, 114, 122, 119, 125, 123, 133, 139, 145, 142, 148, 148, 158, 162, 168, 164, 173, 172, 183, 184, 191, 186, 198, 197, 206, 207, 214, 211, 223, 220
(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 B. (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 B, and the green dots indicate the centers of large empty hexagrams.)
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FORMULA
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The b-file suggests that this sequence has g.f. = f/g, where
f = -4*x^27+2*x^26-2*x^25+2*x^23+2*x^22+6*x^21+7*x^20+16*x^19+19*x^18+20*x^17+
22*x^16+22*x^15+25*x^14+24*x^13+25*x^12+26*x^11+21*x^10+25*x^9+25*x^8+23*x^7+
21*x^6+17*x^5+16*x^4+9*x^3+6*x^2+2*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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