A298031 Coordination sequence of Dual(3.4.6.4) tiling with respect to a tetravalent node.

1, 4, 10, 16, 30, 36, 48, 54, 66, 72, 84, 90, 102, 108, 120, 126, 138, 144, 156, 162, 174, 180, 192, 198, 210, 216, 228, 234, 246, 252, 264, 270, 282, 288, 300, 306, 318, 324, 336, 342, 354, 360, 372, 378, 390, 396, 408, 414, 426, 432, 444, 450, 462, 468, 480, 486, 498, 504, 516, 522, 534, 540

Offset

0,2

Comments

Also known as the mta net.

This is one of the Laves tilings.

In the Ferreol link this is described as the dual to the Diana tiling. - N. J. A. Sloane, May 24 2020

Links

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

Robert Ferreol, Pavage de diane et son dual

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 quadrant 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 >= 4, a(n) = 9*n-6 if n is even, otherwise a(n) = 9*n-9.

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.: -(2*x^6 - 8*x^4 - 3*x^3 - 5*x^2 - 3*x - 1) / ((1 - x)*(1 - x^2)).

a(n) = a(n-1) + a(n-2) - a(n-3) for n>4. - Colin Barker, Jan 25 2018

a(n) = 6*A007494(n-1), n>3. - R. J. Mathar, Jan 29 2018

Maple

f4:=proc(n) local L; L:=[1, 4, 10, 16];
if n<4 then L[n+1] elif (n mod 2) = 0 then 9*n-6 else 9*n-9; fi;
end;
[seq(f4(n), n=0..80)];

Mathematica

Join[{1, 4, 10, 16}, LinearRecurrence[{1, 1, -1}, {30, 36, 48}, 62]] (* Jean-François Alcover, Apr 23 2018 *)

Prog

(PARI) Vec((1 + 3*x + 5*x^2 + 3*x^3 + 8*x^4 - 2*x^6) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018

Crossrefs

Cf. A008574, A298032 (partial sums), A298029 (for a trivalent node), A298033 (hexavalent node).

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: A167346 A307274 A027430 * A027425 A024992 A224966

Adjacent sequences: A298028 A298029 A298030 * A298032 A298033 A298034

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 21 2018; extended with formula, Jan 24 2018.

Status

approved