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A320497
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Coordination sequence of thinnest 5-neighbor packing of the plane with congruent hexagons with respect to a point of type C.
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7
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1, 4, 8, 10, 14, 16, 22, 28, 32, 36, 36, 42, 48, 50, 54, 58, 60, 68, 74, 72, 76, 80, 86, 94, 96, 94, 98, 106, 112, 116, 118, 116, 122, 132, 138, 138, 140, 138, 148, 158, 160, 160, 162, 164, 174, 180, 182, 182, 186, 190, 200, 202, 204, 204, 212, 216, 222, 224
(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 C. (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 C, and the green dots indicate the centers of large empty hexagrams.)
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FORMULA
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G.f.: (1 + 4*x + 8*x^2 + 10*x^3 + 14*x^4 + 16*x^5 + 21*x^6 + 24*x^7 + 24*x^8 + 26*x^9 + 22*x^10 + 26*x^11 + 26*x^12 + 22*x^13 + 22*x^14 + 22*x^15 + 23*x^16 + 22*x^17 + 18*x^18 + 12*x^19 + 8*x^20 + 6*x^21 + 5*x^22 + 2*x^23 - 2*x^24 - 4*x^25) / (x^22 - x^16 - x^6 + 1).
a(n) = a(n-6) + a(n-16) - a(n-22) for n>25.
(End)
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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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