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
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,easy
AUTHOR
N. J. A. Sloane, Jan 22 2018
STATUS
approved