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

0,2

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

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)=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

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 *)

Prog

(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

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.

Sequence in context: A151724 A335795 A315815 * A315816 A315817 A325696

Adjacent sequences: A298023 A298024 A298025 * A298027 A298028 A298029

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 21 2018

Status

approved