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A299272
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Coordination sequence for "flu" 3D uniform tiling formed from tetrahedra, rhombicuboctahedra, and cubes.
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51
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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
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OFFSET
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0,2
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COMMENTS
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The tiling is called "3-RCO-trille" in Conway, Burgiel, Goodman-Strauss, 2008, p. 297. - Felix Fröhlich, Feb 11 2018
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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<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
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CROSSREFS
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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.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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