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%I #21 Feb 12 2021 11:07:49
%S 1,5,12,22,36,56,82,111,144,183,226,272,324,382,442,505,576,653,730,
%T 810,900,996,1090,1187,1296,1411,1522,1636,1764,1898,2026,2157,2304,
%U 2457,2602,2750,2916,3088,3250,3415,3600,3791,3970,4152,4356,4566,4762,4961,5184,5413,5626,5842,6084,6332
%N Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.12.12 2D tiling (cf. A250122).
%D B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #19.
%H Colin Barker, <a href="/A299257/b299257.txt">Table of n, a(n) for n = 0..1000</a>
%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/nets/ttw">The ttw tiling (or net)</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-5,7,-7,5,-3,1).
%F G.f.: (2*x^8 - 4*x^7 + 3*x^6 - 5*x^5 + x^4 - 3*x^3 - x^2 - x - 1)*(x + 1) / ((x - 1)^3*(x^2 + 1)^2).
%F From _Colin Barker_, Feb 09 2018: (Start)
%F a(n) = (4 - (2+8*i)*(-i)^n - (2-8*i)*i^n + i*((-i)^n-i^n)*n + 18*n^2) / 8 for n>2, where i=sqrt(-1).
%F a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7) for n>9.
%F (End)
%F a(n) = 1/2 + 9*n^2/4 + (-1)^floor(n/2)*(A027656(n-1)/2 - A010699(n)/4). - _R. J. Mathar_, Feb 12 2021
%o (PARI) Vec((1 + x)*(1 + x + x^2 + 3*x^3 - x^4 + 5*x^5 - 3*x^6 + 4*x^7 - 2*x^8) / ((1 - x)^3*(1 + x^2)^2) + O(x^60)) \\ _Colin Barker_, Feb 09 2018
%Y Cf. A250122.
%Y Partial sums: A299263.
%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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