A299274 Coordination sequence for "hal" 3D uniform tiling.

1, 4, 9, 18, 30, 47, 69, 91, 125, 160, 191, 238, 282, 331, 391, 448, 508, 582, 650, 709, 790, 877, 964, 1047, 1140, 1253, 1353, 1463, 1560, 1667, 1801, 1908, 2043, 2165, 2297, 2471, 2580, 2737, 2893, 3020, 3202, 3344, 3529, 3686, 3856, 4082, 4205, 4429, 4613, 4765, 5025, 5173, 5410

Offset: 0

Author

N. J. A. Sloane, Feb 10 2018

Keywords

• nonn

Comments

This tiling is the f-tCO-trille tiling, described on page 298 of Symmetries of Things. (Note that there is a typo in the figure caption - tO should be tT.)- Chaim Goodman-Strauss, Feb 13 2018.

Other names for this tiling: In RCSR, HL42, 4/3/c8, 6^2.8^2 F-RD. Wells (page 146), refers to it as the 4-connected Fm3m net or 96(k) or H and L 4_2 (cubic) net, with a reference to Andreini Fig 23.

First 120 terms computed by Davide M. Proserpio using ToposPro.

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.
• B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #21.
• A. F. Wells, Three-dimensional Nets and Polyhedra, Wiley, 1977

Links

• Davide M. Proserpio, Table of n, a(n) for n = 0..120
• V. A. Blatov, A. P. Shevchenko, D. M. Proserpio, Applied Topological Analysis of Crystal Structures with the Program Package ToposPro, Cryst. Growth Des. 2014, 14, 3576-3586.
• Reticular Chemistry Structure Resource (RCSR), The hal tiling (or net)

Crossrefs

See A299275 for partial sums.

The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.

Formula

From N. J. A. Sloane, Feb 13 2018 (Start):

Based on the 120 terms computed from the definition by Davide M. Proserpio, and using gfun, it appears that the g.f. is p(x)/q(x), where p(x) and q(x) are respectively

6*x^43 + 12*x^42 + 26*x^41 + 38*x^40 + 47*x^39 + 45*x^38 + 31*x^37 + 9*x^36 - 14*x^35 - 30*x^34 - 35*x^33 - 10*x^32 + 50*x^31 + 173*x^30 + 368*x^29 + 645*x^28 + 1006*x^27 + 1426*x^26 + 1889*x^25 + 2367*x^24 + 2835*x^23 + 3267*x^22 + 3630*x^21 + 3887*x^20 + 4038*x^19 + 4040*x^18 + 3931*x^17 + 3695*x^16 + 3379*x^15 + 2992*x^14 + 2567*x^13 + 2127*x^12 + 1701*x^11 + 1308*x^10 + 964*x^9 + 680*x^8 + 453*x^7 + 285*x^6 + 166*x^5 + 87*x^4 + 41*x^3 + 16*x^2 + 5*x + 1

and

(x + 1)*(x^2 + 1)*(x^6 + x^3 + 1)*(x^2 + x + 1)^2*(x^4 - x^3 + x^2 - x + 1)^2*(1 - x)^3*(x^4 + x^3 + x^2 + x + 1)^3.

The denominator q(x) can also be written as

(1-x^3)*(1-x^4)*(1-x^5)*(1-x^9)*(1-x^10)^2/((1-x)^3*(1+x)^2).

However, this g.f. is so much more complicated than the g.f.s for any of the other 27 3D uniform tilings, at present I am only willing to state it as a conjecture.

It should not be used to extend the sequence beyond 120 terms. (End)