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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.)