A298036 Coordination sequence of Dual(4.6.12) tiling with respect to a 12-valent node.

1, 12, 12, 36, 24, 60, 36, 84, 48, 108, 60, 132, 72, 156, 84, 180, 96, 204, 108, 228, 120, 252, 132, 276, 144, 300, 156, 324, 168, 348, 180, 372, 192, 396, 204, 420, 216, 444, 228, 468, 240, 492, 252, 516, 264, 540, 276, 564, 288, 588, 300

Offset: 0

Author

N. J. A. Sloane, Jan 22 2018

Keywords

• nonn
• easy

Comments

Conjecture: For n>0, a(n)=6n if n even, otherwise 12n.

The conjecture can easily be shown to be true: The vertices at distance 2k consist of 3k 12-valent and 3k 4-alent vertices, and the vertices at distance 2k+1 consist of 6(k+1) 6-valent and 6(k+1) 4-valent vertices. - Charlie Neder, Apr 22 2019

Links

• Hakan Icoz, Table of n, a(n) for n = 0..20000
• Tom Karzes, Tiling Coordination Sequences
• N. J. A. Sloane, Illustration of initial terms (shows one 60-degree sector of tiling)
• N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A072154, A298037 (partial sums), A298038 (hexavalent node), A298040 (tetravalent node).

Cf. A109043 (a(n)/6), A026741 (a(n)/12).

List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

Programs

Mathematica

LinearRecurrence[{0, 2, 0, -1}, {1, 12, 12, 36, 24}, 100] (* Paolo Xausa, Jul 19 2024 *)

Formula

From Charlie Neder, Apr 22 2019: (Start)

a(n) = 6 * n * (1 + n mod 2), n > 0.

G.f.: (1 + 12*x + 10*x^2 + 12*x^3 + x^4)/(1 - x^2)^2. (End)

Extensions

a(7)-a(50) from Charlie Neder, Apr 22 2019