A298014 Coordination sequence of snub-632 tiling with respect to a trivalent node of type short-short-long.

1, 3, 9, 15, 18, 27, 37, 37, 44, 57, 54, 61, 77, 71, 78, 97, 88, 95, 117, 105, 112, 137, 122, 129, 157, 139, 146, 177, 156, 163, 197, 173, 180, 217, 190, 197, 237, 207, 214, 257, 224, 231, 277, 241, 248, 297, 258, 265, 317, 275, 282, 337, 292, 299, 357, 309, 316, 377, 326, 333, 397, 343, 350, 417, 360

Offset: 0

Author

Chaim Goodman-Strauss and N. J. A. Sloane, Jan 11 2018

Keywords

• nonn
• easy

Comments

The snub-632 tiling in also called the fsz-d net. It is the dual of the 3.3.3.3.6 Archimedean tiling.

This is also called the "6-fold pentille" tiling in Conway, Burgiel, Goodman-Strauss, 2008, p. 288. - Felix Fröhlich, Jan 13 2018

References

• J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5.

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 arXiv:1803.08530.
• 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 (0,0,2,0,0,-1).

Crossrefs

Cf. A298015, A298016.

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.

Programs

Maple

f:=proc(n) local k,r,L; L:=[1,3,9,15,18];
if n<5 then L[n+1]
else k:=floor(n/3); r:=n-3*k;
  if r=0 then 20*k-3 elif r=1 then 17*k+3 else 17*k+10; fi;
fi; end;
[seq(f(n),n=0..80)];

Mathematica

Join[{1, 3, 9, 15, 18}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {27, 37, 37, 44, 57, 54}, 60]] (* Jean-François Alcover, Apr 28 2018 *)

PARI

Vec(-(x+1)*(2*x^9+x^7-5*x^6-3*x^5-6*x^4-6*x^3-7*x^2-2*x-1)/((x-1)^2*(x^2+x+1)^2) + O(x^60)) \\ Colin Barker, Jan 13 2018

Formula

For n >= 5, let k=floor(n/3). Then a(3*k) = 20*k-3, a(3*k+1)=17*k+3, a(3*k+2)=17*k+10.

G.f.: -(x+1)*(2*x^9+x^7-5*x^6-3*x^5-6*x^4-6*x^3-7*x^2-2*x-1)/((x-1)^2*(x^2+x+1)^2).

a(n) = 2*a(n-3) - a(n-6) for n>10. - Colin Barker, Jan 13 2018