A298033 Coordination sequence of the Dual(3.4.6.4) tiling with respect to a hexavalent node.

1, 6, 12, 24, 30, 42, 48, 60, 66, 78, 84, 96, 102, 114, 120, 132, 138, 150, 156, 168, 174, 186, 192, 204, 210, 222, 228, 240, 246, 258, 264, 276, 282, 294, 300, 312, 318, 330, 336, 348, 354, 366, 372, 384, 390, 402, 408, 420, 426, 438, 444, 456, 462, 474, 480, 492, 498, 510, 516, 528, 534, 546, 552

Offset

0,2

Comments

Also known as the mta net.

This is one of the Laves tilings.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.

Tom Karzes, Tiling Coordination Sequences

Reticular Chemistry Structure Resource (RCSR), The mta tiling (or net)

N. J. A. Sloane, The Dual(3.4.6.4) tiling

N. J. A. Sloane, The subgraph H shown in one 60-degree sector of the graph of the 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]

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Formula

Theorem: For n>0, a(n) = 9*n-6 if n is even, a(n) = 9*n-3 if n is odd.

The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.

G.f.: (1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)*(1 - x^2)).

First differences are 1, 5, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, ...

a(n) = a(n-1) + a(n-2) - a(n-3) for n>3. - Colin Barker, Jan 25 2018

a(n) = 6*floor((3n-1)/2) for n > 0. - M. F. Hasler, Jan 11 2022

Maple

f6:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 9*n-6 else 9*n-3; fi; end;
[seq(f6(n), n=0..80)];

Mathematica

Join[{1}, LinearRecurrence[{1, 1, -1}, {6, 12, 24}, 62]] (* Jean-François Alcover, Apr 23 2018 *)

Prog

(PARI) Vec((1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018
(PARI) apply( {A298033(n)=if(n, n*3\/2*6-6, 1)}, [0..66]) \\ M. F. Hasler, Jan 11 2022

Crossrefs

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.

Cf. A008574, A038764 (partial sums), A298029 (coordination sequence for a trivalent node), A298031 (coordination sequence for a tetravalent node).

Sequence in context: A307225 A261476 A119840 * A358507 A358526 A069171

Adjacent sequences: A298030 A298031 A298032 * A298034 A298035 A298036

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 21 2018, corrected Jan 24 2018.

Status

approved