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Coordination sequence T1 for Zeolite Code LTA and RHO.
71

%I #48 Feb 12 2023 16:24:50

%S 1,4,9,17,28,42,60,81,105,132,162,196,233,273,316,362,412,465,521,580,

%T 642,708,777,849,924,1002,1084,1169,1257,1348,1442,1540,1641,1745,

%U 1852,1962,2076,2193,2313,2436,2562,2692,2825,2961,3100,3242,3388,3537,3689

%N Coordination sequence T1 for Zeolite Code LTA and RHO.

%C Also, growth series for the affine Coxeter (or Weyl) groups B_3. - _N. J. A. Sloane_, Jan 11 2016

%C Also, coordination sequence for "rho" 3D uniform tiling. - _N. J. A. Sloane_, Feb 10 2018

%D N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10b, page 231, W_a(t).

%D B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tilings #25 and 27.

%D W. M. Meier, D. H. Olson and Ch. Baerlocher, Atlas of Zeolite Structure Types, 4th Ed., Elsevier, 1996.

%H R. W. Grosse-Kunstleve, <a href="/A008137/b008137.txt">Table of n, a(n) for n = 0..1000</a>

%H R. W. Grosse-Kunstleve, <a href="/A005897/a005897.html">Coordination Sequences and Encyclopedia of Integer Sequences</a>

%H R. W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, <a href="http://neilsloane.com/doc/ac96cs/">Algebraic Description of Coordination Sequences and Exact Topological Densities for Zeolites</a>, Acta Cryst., A52 (1996), pp. <a href="http://dx.doi.org/10.1107/S0108767396007519">879-889</a>.

%H Sean A. Irvine, <a href="/A008000/a008000_1.pdf">Generating Functions for Coordination Sequences of Zeolites, after Grosse-Kunstleve, Brunner, and Sloane</a>

%H International Zeolite Association, <a href="http://www.iza-structure.org/databases/">Database of Zeolite Structures</a>

%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/nets/lta">The lta tiling (or net)</a>

%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/nets/rho">The rho tiling (or net)</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,1,-2,1).

%F a(5*m+k) = 40*m^2 + 16*k*m + one of 5 numbers depending on k, 0 <= k < 5 (_N. J. A. Sloane_).

%F G.f.: (1-x^2)*(1-x^4)*(1-x^6)/((1-x)^4*(1-x^3)*(1-x^5)). This can also be written as (x+1)^3*(x^2+1)*(x^2-x+1)/((1-x)^3*(x^4+x^3+x^2+x+1)). - _N. J. A. Sloane_, Feb 10 2018

%F a(n) = 12/5 - 0^n + (8/5)*n^2 - (1/25)*(5+sqrt(5))*cos(2*Pi*n/5) - (1/25)*(5-sqrt(5))*cos(4*Pi*n/5). - _Eric Simon Jacob_, Feb 12 2023

%p (1-x^2)*(1-x^4)*(1-x^6)/((1-x)^4*(1-x^3)*(1-x^5));

%p seq(coeff(series(%,x,n+1),x,n), n=0..48);

%Y The growth series for the finite Coxeter (or Weyl) groups B_2 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858. These are all rows of A128084. The growth series for the affine Coxeter (or Weyl) groups B_2 through B_12 are A008576, A008137, A267167-A267175.

%Y For partial sums see A299276.

%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,easy

%O 0,2

%A _Ralf W. Grosse-Kunstleve_