A298035 Coordination sequence of Dual(3.12.12) tiling with respect to a trivalent node.

1, 3, 21, 39, 57, 75, 93, 111, 129, 147, 165, 183, 201, 219, 237, 255, 273, 291, 309, 327, 345, 363, 381, 399, 417, 435, 453, 471, 489, 507, 525, 543, 561, 579, 597, 615, 633, 651, 669, 687, 705, 723, 741, 759, 777, 795, 813, 831, 849, 867, 885, 903, 921, 939, 957, 975, 993, 1011, 1029, 1047, 1065

Offset

0,2

Comments

This tiling is sometimes called the triakis triangular tiling.

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 on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.

Tom Karzes, Tiling Coordination Sequences

N. J. A. Sloane, Illustration of initial terms (shows one 120-degree sector of graph).

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 (2,-1).

Formula

Theorem: a(0)=1; thereafter a(n) = 18*n-15. [Proof: Use the "coloring book" method described in the Goodman-Strauss & Sloane article.]

From Colin Barker, Jan 22 2018: (Start)

G.f.: (1 + x + 16*x^2) / (1 - x)^2.

a(n) = 2*a(n-1) - a(n-2) for n>2.

(End)

Maple

f3:=proc(n) if n=0 then 1 else 18*n-15; fi; end;
[seq(f3(n), n=0..80)];

Prog

(PARI) Vec((1 + x + 16*x^2) / (1 - x)^2 + O(x^60)) \\ Colin Barker, Jan 22 2018

Crossrefs

Cf. A019557 (12-valent node), A016790 (partial sums, provided its offset is changed).

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: A191763 A227241 A076169 * A178082 A056387 A056377

Adjacent sequences: A298032 A298033 A298034 * A298036 A298037 A298038

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 22 2018

Status

approved