A298015 Coordination sequence of snub-632 tiling with respect to a trivalent node of type short-short-short.

1, 3, 6, 15, 21, 18, 33, 48, 30, 51, 72, 42, 69, 96, 54, 87, 120, 66, 105, 144, 78, 123, 168, 90, 141, 192, 102, 159, 216, 114, 177, 240, 126, 195, 264, 138, 213, 288, 150, 231, 312, 162, 249, 336, 174, 267, 360, 186, 285, 384, 198, 303, 408, 210, 321, 432, 222, 339, 456, 234, 357, 480, 246, 375

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. [Warning: there is an error in Eq. 8(b), the a(4) term should be changed from 24 to 21. With that correction Theorem then still holds. - _N. J. A. Sloane_, Apr 01 2020]
• Tom Karzes, Illustration of a(0) to a(4) [Key: n, a(n), color: 0, 1, green; 1, 3, red; 2, 6, blue; 3, 15, purple; 4, 21, beige.]
• 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. A298014, 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.

Formula

For n >= 6, let k=floor(n/3), so k >= 2. Then a(3*k) = 18*k-3, a(3*k+1)=24*k, a(3*k+2)=12*k+6. [Corrected by N. J. A. Sloane, Apr 01 2020]

a(n) = 2*a(n-3) - a(n-6) for n>=11. [Corrected by N. J. A. Sloane, Apr 01 2020]

G.f.: -(3*x^10-9*x^7-4*x^6-6*x^5-15*x^4-13*x^3-6*x^2-3*x-1)/(x^6-2*x^3+1). [Corrected by N. J. A. Sloane, Apr 01 2020]

Extensions

a(4) corrected by Tom Karzes. I corrected the b-file and the formulas and deleted the programs. - N. J. A. Sloane, Apr 01 2020