A298038 Coordination sequence of Dual(4.6.12) tiling with respect to a hexavalent node.

1, 6, 24, 18, 48, 30, 72, 42, 96, 54, 120, 66, 144, 78, 168, 90, 192, 102, 216, 114, 240, 126, 264, 138, 288, 150, 312, 162, 336, 174, 360, 186, 384, 198, 408, 210, 432, 222, 456, 234, 480, 246, 504, 258, 528, 270, 552, 282, 576, 294, 600, 306, 624, 318, 648

Offset: 0

Author

N. J. A. Sloane, Jan 22 2018

Keywords

• nonn

Comments

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

From Keagan Boyce, May 18 2024: (Start)

It appears that

a(n) = (3*n)*(3+(-1)^n) for n > 0,

which would imply that for all even n > 0,

a(n) = (3*n)*(3+(1)) = (3*n)*(4) = 12*n,

and for all odd n > 0,

a(n) = (3*n)*(3+(-1)) = (3*n)*(2) = 6*n. (End)

Links

• 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].

Crossrefs

Cf. A072154, A298039 (partial sums), A298036 (12-valent node), A298040 (tetravalent node).

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.

Formula

Conjectures from Colin Barker, Apr 03 2020: (Start)

G.f.: (1 + 6*x + 22*x^2 + 6*x^3 + x^4) / ((1 - x)^2*(1 + x)^2).

a(n) = 2*a(n-2) - a(n-4) for n > 4. (End)

Extensions

Terms a(8)-a(54) added by Tom Karzes, Apr 01 2020