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A298029 Coordination sequence of Dual(3.4.6.4) tiling with respect to a trivalent node. 22
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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>5. - 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

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 *)

PROG

(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

CROSSREFS

Cf. A008574, A298030 (partial sums), A298031 (for a tetravalent 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: A171370 A061776 A341316 * A327328 A074899 A318872

Adjacent sequences:  A298026 A298027 A298028 * A298030 A298031 A298032

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Jan 21 2018

STATUS

approved

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Last modified June 17 18:28 EDT 2021. Contains 345085 sequences. (Running on oeis4.)