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%I #29 Feb 18 2024 04:31:39
%S 1,5,13,26,45,69,98,133,173,218,269,325,386,453,525,602,685,773,866,
%T 965,1069,1178,1293,1413,1538,1669,1805,1946,2093,2245,2402,2565,2733,
%U 2906,3085,3269,3458,3653,3853,4058,4269,4485,4706,4933,5165,5402,5645,5893,6146,6405,6669,6938,7213,7493
%N Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.8.8 2D tiling (cf. A008576).
%H Colin Barker, <a href="/A299259/b299259.txt">Table of n, a(n) for n = 0..1000</a>
%H B. Grünbaum, <a href="https://faculty.washington.edu/moishe/branko/BG199.Uniform%20Tilings%20of%203-Space.pdf">Uniform tilings of 3-space, Geombinatorics</a>, 4 (1994), 49-56. See tiling #24.
%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/nets/fee">The fee tiling (or net)</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F G.f.: (x + 1)^3*(x^2 + 1) / ((1 - x)^3*(x^2 + x + 1)).
%F a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>5. - _Colin Barker_, Feb 09 2018
%F a(n) = (4*(5 + 6*n^2) + A061347(n))/9 for n > 0. - _Stefano Spezia_, Feb 17 2024
%t CoefficientList[Series[(x+1)^3*(x^2+1)/((1-x)^3*(x^2+x+1)), {x, 0, 50}], x] (* _G. C. Greubel_, Feb 20 2018 *)
%o (PARI) Vec((1 + x)^3*(1 + x^2) / ((1 - x)^3*(1 + x + x^2)) + O(x^60)) \\ _Colin Barker_, Feb 09 2018
%o (Magma) I:=[13, 26, 45, 69, 98]; [1,5] cat [n le 5 select I[n] else 2*Self(n-1) - Self(n-2) + Self(n-3) - 2*Self(n-4) + Self(n-5): n in [1..30]]; // _G. C. Greubel_, Feb 20 2018
%Y Cf. A008576, A061347.
%Y Partial sums give A299265.
%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 _N. J. A. Sloane_, Feb 07 2018
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