OFFSET
0,2
COMMENTS
Also known as the mta net.
This is one of the Laves tilings.
In the Ferreol link this is described as the dual to the Diana tiling. - N. J. A. Sloane, May 24 2020
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, The subgraph H shown in one quadrant of the graph of the 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 (1,1,-1).
FORMULA
Theorem: For n >= 4, a(n) = 9*n-6 if n is even, otherwise a(n) = 9*n-9.
The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.
G.f.: -(2*x^6 - 8*x^4 - 3*x^3 - 5*x^2 - 3*x - 1) / ((1 - x)*(1 - x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>4. - Colin Barker, Jan 25 2018
a(n) = 6*A007494(n-1), n>3. - R. J. Mathar, Jan 29 2018
MAPLE
f4:=proc(n) local L; L:=[1, 4, 10, 16];
if n<4 then L[n+1] elif (n mod 2) = 0 then 9*n-6 else 9*n-9; fi;
end;
[seq(f4(n), n=0..80)];
MATHEMATICA
Join[{1, 4, 10, 16}, LinearRecurrence[{1, 1, -1}, {30, 36, 48}, 62]] (* Jean-François Alcover, Apr 23 2018 *)
PROG
(PARI) Vec((1 + 3*x + 5*x^2 + 3*x^3 + 8*x^4 - 2*x^6) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 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 21 2018; extended with formula, Jan 24 2018.
STATUS
approved