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 A316320 Coordination sequence for a hexavalent node in a chamfered version of the 3^6 triangular tiling of the plane. 2
 1, 6, 15, 27, 39, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. REFERENCES J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., 1993. See Fig. 7.2, page 199. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Rémy Sigrist, Illustration of initial terms N. J. A. Sloane, The graph of the tiling. (The red dots indicate the nodes of the sublattice theta*E.) Index entries for linear recurrences with constant coefficients, signature (2,-1). FORMULA a(n) = 12*n-9 = A017557(n-1) for n > 1. From Colin Barker, Mar 11 2020: (Start) G.f.: (1 + 3*x)*(1 + x + x^2) / (1 - x)^2. a(n) = 2*a(n-1) - a(n-2) for n>3. (End) PROG (PARI) Vec((1 + 3*x)*(1 + x + x^2) / (1 - x)^2 + O(x^50)) \\ Colin Barker, Mar 11 2020 CROSSREFS See A316319 for trivalent node. See A250120 for links to thousands of other coordination sequences. Cf. A017557, A008486. Sequence in context: A240948 A072257 A227952 * A140091 A255605 A171972 Adjacent sequences:  A316317 A316318 A316319 * A316321 A316322 A316323 KEYWORD nonn,easy AUTHOR Rémy Sigrist and N. J. A. Sloane, Jul 01 2018 EXTENSIONS Terms a(15) and beyond from Andrey Zabolotskiy, Sep 30 2019 STATUS approved

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Last modified September 29 09:40 EDT 2020. Contains 337428 sequences. (Running on oeis4.)