

A298036


Coordination sequence of Dual(4.6.12) tiling with respect to a 12valent node.


23



1, 12, 12, 36, 24, 60, 36, 84, 48, 108, 60, 132, 72, 156, 84, 180, 96, 204, 108, 228, 120, 252, 132, 276, 144, 300, 156, 324, 168, 348, 180, 372, 192, 396, 204, 420, 216, 444, 228, 468, 240, 492, 252, 516, 264, 540, 276, 564, 288, 588, 300
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OFFSET

0,2


COMMENTS

Conjecture: For n>0, a(n)=6n if n even, otherwise 12n.
The conjecture can easily be shown to be true: The vertices at distance 2k consist of 3k 12valent and 3k 4alent vertices, and the vertices at distance 2k+1 consist of 6(k+1) 6valent and 6(k+1) 4valent vertices.  Charlie Neder, Apr 22 2019


LINKS

Hakan Icoz, Table of n, a(n) for n = 0..20000
Tom Karzes, Tiling Coordination Sequences
N. J. A. Sloane, Illustration of initial terms (shows one 60degree sector of tiling)
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of GrünbaumShephard 1987 with Anumbers added and in some cases the name in the RCSR database]


FORMULA

From Charlie Neder, Apr 22 2019: (Start)
a(n) = 6 * n * (1 + n mod 2), n > 0.
G.f.: (1 + 12*x + 10*x^2 + 12*x^3 + x^4)/(1  x^2)^2. (End)


CROSSREFS

Cf. A072154, A298037 (partial sums), A298038 (hexavalent node), A298040 (tetravalent node).
Cf. A109043 (a(n)/6), A026741 (a(n)/12).
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: A070710 A048759 A303646 * A119877 A307842 A147833
Adjacent sequences: A298033 A298034 A298035 * A298037 A298038 A298039


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Jan 22 2018


EXTENSIONS

a(7)a(50) from Charlie Neder, Apr 22 2019


STATUS

approved



