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%I #52 Jan 19 2020 18:13:17
%S 1,4,9,18,30,47,69,91,125,160,191,238,282,331,391,448,508,582,650,709,
%T 790,877,964,1047,1140,1253,1353,1463,1560,1667,1801,1908,2043,2165,
%U 2297,2471,2580,2737,2893,3020,3202,3344,3529,3686,3856,4082,4205,4429,4613,4765,5025,5173,5410
%N Coordination sequence for "hal" 3D uniform tiling.
%C 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.
%C 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.
%C First 120 terms computed by _Davide M. Proserpio_ using ToposPro.
%D J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5.
%D B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #21.
%D A. F. Wells, Three-dimensional Nets and Polyhedra, Wiley, 1977
%H Davide M. Proserpio, <a href="/A299274/b299274.txt">Table of n, a(n) for n = 0..120</a>
%H V. A. Blatov, A. P. Shevchenko, D. M. Proserpio, <a href="http://pubs.acs.org/doi/pdf/10.1021/cg500498k">Applied Topological Analysis of Crystal Structures with the Program Package ToposPro</a>, Cryst. Growth Des. 2014, 14, 3576-3586.
%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/nets/hal">The hal tiling (or net)</a>
%F From _N. J. A. Sloane_, Feb 13 2018 (Start):
%F 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
%F 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
%F and
%F (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.
%F The denominator q(x) can also be written as
%F (1-x^3)*(1-x^4)*(1-x^5)*(1-x^9)*(1-x^10)^2/((1-x)^3*(1+x)^2).
%F 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.
%F It should not be used to extend the sequence beyond 120 terms. (End)
%Y See A299275 for partial sums.
%Y 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.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Feb 10 2018
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