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%I #34 Sep 08 2022 08:46:20
%S 1,6,18,37,63,99,142,189,249,317,384,468,562,648,756,877,981,1113,
%T 1262,1383,1539,1717,1854,2034,2242,2394,2598,2837,3003,3231,3502,
%U 3681,3933,4237,4428,4704,5042,5244,5544,5917,6129,6453,6862,7083,7431,7877,8106,8478,8962,9198
%N Coordination sequence for "flu" 3D uniform tiling formed from tetrahedra, rhombicuboctahedra, and cubes.
%C First 20 terms computed by _Davide M. Proserpio_ using ToposPro.
%C The tiling is called "3-RCO-trille" in Conway, Burgiel, Goodman-Strauss, 2008, p. 297. - _Felix Fröhlich_, Feb 11 2018
%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 #5.
%H G. C. Greubel, <a href="/A299272/b299272.txt">Table of n, a(n) for n = 0..5000</a>
%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/nets/flu">The flu tiling (or net)</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycomb#Runcic_cubic_honeycomb">Tetrahedral-octahedral honeycomb - Runcic cubic honeycomb</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,3,0,0,-3,0,0,1).
%F Conjectures from _Colin Barker_, Feb 11 2018: (Start)
%F G.f.: (1 + x)^3*(1 + x^2)*(1 + 3*x + 5*x^2 + 3*x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)^3).
%F a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n>9.
%F (End)
%F G.f.: (x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3 / (1-x^3)^3. - _N. J. A. Sloane_, Feb 12 2018 (This confirms my conjecture from Feb 10 2018 and the above conjecture from _Colin Barker_.)
%F a(n) = (60 + 104*n^2 + (n^2 - 6)*cos(2*n*Pi/3) - 3*sqrt(3)*n*sin(2*n*Pi/3))/27 for n > 0. - _Stefano Spezia_, Jan 23 2022
%t CoefficientList[Series[(x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3/(1-x^3)^3, {x, 0, 50}], x] (* _G. C. Greubel_, Feb 20 2018 *)
%o (PARI) x='x+O('x^30); Vec((x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3/(1-x^3)^3) \\ _G. C. Greubel_, Feb 20 2018
%o (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3/(1-x^3)^3)) // _G. C. Greubel_, Feb 20 2018
%Y See A299273 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,easy
%O 0,2
%A _N. J. A. Sloane_, Feb 10 2018
%E a(21)-a(40) from _Davide M. Proserpio_, Feb 12 2018
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