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

0,2

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).

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

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 *)

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.

Sequence in context: A114237 A060035 A165988 * A215842 A018876 A038230

Adjacent sequences: A298025 A298026 A298027 * A298029 A298030 A298031

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 21 2018

Status

approved