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A316319
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Coordination sequence for a trivalent node in a chamfered version of the 3^6 triangular tiling of the plane.
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2
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1, 3, 7, 14, 25, 38, 51, 63, 75, 87, 99, 111, 123, 135, 147, 159, 171, 183, 195, 207, 219, 231, 243, 255, 267, 279, 291, 303, 315, 327, 339, 351, 363, 375, 387, 399, 411, 423, 435, 447, 459, 471, 483, 495, 507, 519, 531, 543, 555, 567, 579, 591, 603, 615, 627, 639
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OFFSET
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0,2
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COMMENTS
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Let E denote the lattice of Eisenstein integers u + v*w in the plane, with each point joined to its six neighbors. Here u and v are ordinary integers and w = (-1+sqrt(-3))/2 is a complex cube root of unity. Let theta = w - w^2 = sqrt(-3). Then theta*E is a sublattice of E of index 3 (Conway-Sloane, Fig. 7.2). The tiling considered in this sequence is obtained by replacing each node in theta*E by a small hexagon.
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., 1993. See Fig. 7.2, page 199.
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LINKS
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FORMULA
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a(n) = 12*n-21 = A017557(n-2) for n > 5.
G.f.: (1 + x + x^2)*(1 + x^2 + 2*x^3 + x^4 - x^5) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>7.
(End)
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PROG
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(PARI) Vec((1 + x + x^2)*(1 + x^2 + 2*x^3 + x^4 - x^5) / (1 - x)^2 + O(x^50)) \\ Colin Barker, Mar 11 2020
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CROSSREFS
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See A250120 for links to thousands of other coordination sequences.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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