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A320498
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Coordination sequence of thinnest 5-neighbor packing of the plane with congruent hexagons with respect to a point of type D.
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7
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1, 4, 8, 12, 13, 17, 23, 28, 32, 35, 37, 42, 48, 52, 53, 57, 62, 69, 72, 74, 75, 81, 87, 93, 95, 95, 99, 106, 112, 117, 116, 117, 124, 133, 136, 139, 138, 141, 149, 157, 159, 160, 162, 166, 174, 181, 180, 182, 187, 193, 198, 203, 202, 206, 212, 217, 221, 224
(list;
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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 D. (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 D, and the green dots indicate the centers of large empty hexagrams.)
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FORMULA
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G.f.: (1 + 3*x + 5*x^2 + 7*x^3 + 6*x^4 + 11*x^5 + 11*x^6 + 13*x^7 + 11*x^8 + 12*x^9 + 12*x^10 + 13*x^11 + 12*x^12 + 12*x^13 + 9*x^14 + 13*x^15 + 11*x^16 + 12*x^17 + 4*x^18 + 6*x^19 + 3*x^20 + 4*x^21 - x^22 + x^23 - 2*x^24) / (x^21 - x^20 + x^19 - x^18 + x^17 - x^16 - x^5 + x^4 - x^3 + x^2 - x + 1).
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) + a(n-16) - a(n-17) + a(n-18) - a(n-19) + a(n-20) - a(n-21) for n>24.
(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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