A298026 Coordination sequence of Dual(3.6.3.6) tiling with respect to a hexavalent node.

1, 6, 6, 18, 12, 30, 18, 42, 24, 54, 30, 66, 36, 78, 42, 90, 48, 102, 54, 114, 60, 126, 66, 138, 72, 150, 78, 162, 84, 174, 90, 186, 96, 198, 102, 210, 108, 222, 114, 234, 120, 246, 126, 258, 132, 270, 138, 282, 144, 294, 150, 306, 156, 318, 162, 330, 168, 342, 174, 354, 180, 366, 186, 378, 192, 390

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
• 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, A298027 (partial sums), A298028 (trivalent point).

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

f6:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 3*n else 6*n; fi; end;
[seq(f6(n),n=0..80)];

Mathematica

Join[{1}, LinearRecurrence[{0, 2, 0, -1}, {6, 6, 18, 12}, 80]] (* Jean-François Alcover, Mar 23 2020 *)

PARI

Vec((1 + 6*x + 4*x^2 + 6*x^3 + x^4) / ((1 - x)^2*(1 + x)^2) + O(x^60)) \\ Colin Barker, Jan 22 2018

Formula

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

G.f.: 1 + 6*x*(1+x+x^2)/(1-x^2)^2. - Robert Israel, Jan 21 2018

From Colin Barker, Jan 22 2018: (Start)

a(n) = 3*n for n>0 and even.

a(n) = 6*n for n odd.

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

(End)

a(n) = 6*A026741(n), n>0. - R. J. Mathar, Jan 29 2018