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

Author

Rémy Sigrist and N. J. A. Sloane, Jul 01 2018

Keywords

• nonn
• easy

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).

Crossrefs

See A316319 for trivalent node.

See A250120 for links to thousands of other coordination sequences.

Cf. A017557, A008486.

Programs

PARI

Vec((1 + 3*x)*(1 + x + x^2) / (1 - x)^2 + O(x^50)) \\ Colin Barker, Mar 11 2020

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)

Extensions

Terms a(15) and beyond from Andrey Zabolotskiy, Sep 30 2019