A316320 Coordination sequence for a hexavalent node in a chamfered version of the 3^6 triangular tiling of the plane.

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

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: A072257 A227952 A373384 * 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