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 12-valent and 3k 4-alent vertices, and the vertices at distance 2k+1 consist of 6(k+1) 6-valent and 6(k+1) 4-valent 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 60-degree sector of 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 (0,2,0,-1).
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)
MATHEMATICA
LinearRecurrence[{0, 2, 0, -1}, {1, 12, 12, 36, 24}, 100] (* Paolo Xausa, Jul 19 2024 *)
CROSSREFS
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.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 22 2018
EXTENSIONS
a(7)-a(50) from Charlie Neder, Apr 22 2019
STATUS
approved