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A299274
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Coordination sequence for "hal" 3D uniform tiling.
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51
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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
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OFFSET
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0,2
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COMMENTS
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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.
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REFERENCES
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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
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LINKS
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FORMULA
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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)
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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