A298029 Coordination sequence of Dual(3.4.6.4) tiling with respect to a trivalent node.

1, 3, 6, 12, 18, 33, 39, 51, 57, 69, 75, 87, 93, 105, 111, 123, 129, 141, 147, 159, 165, 177, 183, 195, 201, 213, 219, 231, 237, 249, 255, 267, 273, 285, 291, 303, 309, 321, 327, 339, 345, 357, 363, 375, 381, 393, 399, 411, 417, 429, 435, 447, 453, 465, 471, 483, 489, 501, 507, 519, 525, 537, 543, 555

Offset: 0

Author

N. J. A. Sloane, Jan 21 2018

Keywords

• nonn
• easy

Comments

Also known as the deltoidal trihexagonal tiling, or the mta net.

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

This is one of the Laves tilings.

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, 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]
• Wikipedia, Laves tilings
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A007310, A008574, A298030 (partial sums), A298031 (for a tetravalent node), A298033 (hexavalent node), A306771.

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

Mathematica

Join[{1, 3, 6, 12, 18}, LinearRecurrence[{1, 1, -1}, {33, 39, 51}, 60]] (* Jean-François Alcover, Jan 07 2019 *)
Join[{1,3,6,12,18},Table[If[EvenQ[n],9n-15,9n-12],{n,5,70}]] (* Harvey P. Dale, Aug 25 2019 *)

PARI

Vec((1 + 2*x + 2*x^2 + 4*x^3 + 3*x^4 + 9*x^5 - 3*x^7) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018

Formula

Theorem: For n >= 5, if n is even then a(n) = 9*n-15, otherwise a(n) = 9*n-12. The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. - N. J. A. Sloane, Jan 24 2018

G.f.: -(3*x^7 - 9*x^5 - 3*x^4 - 4*x^3 - 2*x^2 - 2*x - 1)/((1 - x)*(1 - x^2)).

a(n) = a(n-1) + a(n-2) - a(n-3) for n>7. - Colin Barker, Jan 25 2018

a(n) = (3/2)*(6*n - (-1)^n - 9) for n>4. - Bruno Berselli, Jan 25 2018

a(n) = 3*A007310(n-1), n>4. - R. J. Mathar, Jan 29 2018