A298028 Coordination sequence of Dual(3.6.3.6) tiling with respect to a trivalent node.

1, 3, 12, 9, 24, 15, 36, 21, 48, 27, 60, 33, 72, 39, 84, 45, 96, 51, 108, 57, 120, 63, 132, 69, 144, 75, 156, 81, 168, 87, 180, 93, 192, 99, 204, 105, 216, 111, 228, 117, 240, 123, 252, 129, 264, 135, 276, 141, 288, 147, 300, 153, 312, 159, 324, 165, 336, 171, 348, 177, 360, 183, 372, 189, 384, 195

Offset: 0

Author

N. J. A. Sloane, Jan 21 2018

Keywords

• nonn
• easy

Comments

Also known as the kgd net.

This is one of the Laves tilings.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Tom Karzes, Tiling Coordination Sequences
• Reticular Chemistry Structure Resource (RCSR), The kgd tiling (or net)
• 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. A008579, A135556 (partial sums), A298026 (trivalent point).

If the initial 1 is changed to 0 we get A165988 (but we need both sequences, just as we have both A008574 and A008586).

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

Maple

f3:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 6*n else 3*n; fi; end;
[seq(f3(n),n=0..80)];

Mathematica

Join[{1}, LinearRecurrence[{0, 2, 0, -1}, {3, 12, 9, 24}, 80]] (* Jean-François Alcover, Mar 23 2020 *)

Formula

a(0)=1; a(2*k) = 12*k, a(2*k+1) = 6*k+3.

G.f.: 1 + 3*x*(x^2+4*x+1)/(1-x^2)^2. - Robert Israel, Jan 21 2018

a(n) = 3*A022998(n), n>0. - R. J. Mathar, Jan 29 2018