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A320494
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Coordination sequence of thinnest 5-neighbor packing of the plane with congruent triangles with respect to a tetravalent point.
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8
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1, 4, 11, 15, 22, 28, 32, 39, 45, 48, 56, 62, 65, 73, 78, 82, 90, 95, 99, 106, 112, 116, 123, 129, 132, 140, 146, 149, 157, 162, 166, 174, 179, 183, 190, 196, 200, 207, 213, 216, 224, 230, 233, 241, 246, 250, 258, 263, 267, 274, 280, 284, 291, 297, 300, 308
(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 triangle has a point in common with exactly five other triangles.
This packing is actually the thinnest 5-neighbor packing in the plane using any congruent convex polygons.
More formally, this sequence is the coordination sequence of the vertex-edge graph of the packing with respect to a tetavalent vertex. The base vertex is marked "B" in the figure (it is the midpoint of an edge of the large empty triangle).
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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. 19.1a, page 32. There is an error in the figure: the triangle 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 triangles are shaded, the base point is marked B, and the green dots indicate the centers of large empty triangles.)
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FORMULA
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G.f.: (1 + 4*x + 11*x^2 + 14*x^3 + 18*x^4 + 16*x^5 + 13*x^6 + 6*x^7 + 3*x^8 - 2*x^9) / ((1 - x)^2*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-3) + a(n-5) - a(n-8) for n>9.
(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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