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A299266 Coordination sequence for "cab" 3D uniform tiling formed from octahedra and truncated cubes.

%I A299266 #34 Sep 08 2022 08:46:20
%S A299266 1,5,9,22,37,57,82,117,145,178,229,281,322,377,445,514,577,645,730,
%T A299266 825,901,982,1093,1205,1294,1397,1525,1654,1765,1881,2026,2181,2305,
%U A299266 2434,2605,2777,2914,3065,3253,3442,3601,3765,3970,4185,4357,4534,4765,4997,5182,5381,5629,5878,6085,6297,6562,6837
%N A299266 Coordination sequence for "cab" 3D uniform tiling formed from octahedra and truncated cubes.
%C A299266 First 20 terms computed by _Davide M. Proserpio_ using ToposPro.
%D A299266 B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #8.
%H A299266 Colin Barker, <a href="/A299266/b299266.txt">Table of n, a(n) for n = 0..1000</a>
%H A299266 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 A299266 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/nets/cab">The cab tiling (or net)</a>
%H A299266 Davide M. Proserpio, <a href="/A299266/a299266.txt">Summary of the 28 uniform 3D tilings and their coordination sequences (produced by ToposPro)</a>
%H A299266 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,2,0,0,0,-2,1,-1,1).
%F A299266 G.f.: (4*x^12 -4*x^11 +x^10 +5*x^8 +20*x^7 +18*x^6 +24*x^5 +14*x^4 +16*x^3 +5*x^2 +4*x +1)/((1-x)*(1-x^2)*(1-x^3)*(1+x^2)^2). - _N. J. A. Sloane_, Feb 12 2018
%F A299266 a(n) = a(n-1) - a(n-2) + 2*a(n-3) - 2*a(n-7) + a(n-8) - a(n-9) + a(n-10) for n>12. - _Colin Barker_, Feb 15 2018
%t A299266 CoefficientList[Series[(4*x^12-4*x^11+x^10+5*x^8+20*x^7+18*x^6+24*x^5 +14*x^4+16*x^3+5*x^2+4*x+1)/((1-x)*(1-x^2)*(1-x^3)*(1+x^2)^2), {x,0, 50}], x] (* _G. C. Greubel_, Feb 20 2018 *)
%o A299266 (PARI) Vec((1 + 4*x + 5*x^2 + 16*x^3 + 14*x^4 + 24*x^5 + 18*x^6 + 20*x^7 + 5*x^8 + x^10 - 4*x^11 + 4*x^12) / ((1 - x)^3*(1 + x)*(1 + x^2)^2*(1 + x + x^2)) + O(x^60)) \\ _Colin Barker_, Feb 15 2018
%o A299266 (Magma) I:=[22, 37, 57, 82, 117, 145, 178,229, 281,322]; [1,5,9] cat [n le 10 select I[n] else Self(n-1) -Self(n-2) +2*Self(n-3)-2*Self(n-7)+Self(n-8)-Self(n-9) + Self(n-10): n in [1..30]]; // _G. C. Greubel_, Feb 20 2018
%Y A299266 See A299267 for partial sums.
%Y A299266 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 A299266 nonn,easy
%O A299266 0,2
%A A299266 _N. J. A. Sloane_, Feb 07 2018
%E A299266 a(21)-a(40) from _Davide M. Proserpio_, Feb 12 2018