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 A299272 Coordination sequence for "flu" 3D uniform tiling formed from tetrahedra, rhombicuboctahedra, and cubes. 51
 1, 6, 18, 37, 63, 99, 142, 189, 249, 317, 384, 468, 562, 648, 756, 877, 981, 1113, 1262, 1383, 1539, 1717, 1854, 2034, 2242, 2394, 2598, 2837, 3003, 3231, 3502, 3681, 3933, 4237, 4428, 4704, 5042, 5244, 5544, 5917, 6129, 6453, 6862, 7083, 7431, 7877, 8106, 8478, 8962, 9198 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS First 20 terms computed by Davide M. Proserpio using ToposPro. The tiling is called "3-RCO-trille" in Conway, Burgiel, Goodman-Strauss, 2008, p. 297. - Felix Fröhlich, Feb 11 2018 REFERENCES J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5. B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #5. LINKS G. C. Greubel, Table of n, a(n) for n = 0..5000 Reticular Chemistry Structure Resource (RCSR), The flu tiling (or net) Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1). FORMULA Conjectures from Colin Barker, Feb 11 2018: (Start) 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). a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n>9. (End) 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.) 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 MATHEMATICA 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 *) PROG (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 (Magma) Q:=Rationals(); R:=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 CROSSREFS See A299273 for partial sums. 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. Sequence in context: A180438 A202366 A185223 * A101853 A132432 A005899 Adjacent sequences: A299269 A299270 A299271 * A299273 A299274 A299275 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Feb 10 2018 EXTENSIONS a(21)-a(40) from Davide M. Proserpio, Feb 12 2018 STATUS approved

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Last modified March 31 15:31 EDT 2023. Contains 361668 sequences. (Running on oeis4.)