A316319 Coordination sequence for a trivalent node in a chamfered version of the 3^6 triangular tiling of the plane.

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

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-21 = A017557(n-2) for n > 5.

From Colin Barker, Mar 11 2020: (Start)

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)

E.g.f.: 22 + 12*x + 2*x^2 - x^3/6 - x^4/12 - x^5/120 + 3*exp(x)*(4*x - 7). - Stefano Spezia, Jul 10 2026

Mathematica

Join[{1, 3, 7, 14, 25, 38}, 12*Range[6, 100] - 21] (* Paolo Xausa, Jul 10 2026 *)

Prog

(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

Crossrefs

See A316320 for hexavalent node.

See A250120 for links to thousands of other coordination sequences.

Cf. A017557.

Sequence in context: A060999 A089187 A333980 * A368205 A179178 A171973

Adjacent sequences: A316316 A316317 A316318 * A316320 A316321 A316322

Keyword

nonn,easy,changed

Author

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

Extensions

Terms a(16) and beyond from Andrei Zabolotskii, Sep 30 2019

Status

approved