A299287 Coordination sequence for "tcd" 3D uniform tiling.
1, 10, 33, 72, 126, 196, 281, 382, 498, 630, 777, 940, 1118, 1312, 1521, 1746, 1986, 2242, 2513, 2800, 3102, 3420, 3753, 4102, 4466, 4846, 5241, 5652, 6078, 6520, 6977, 7450, 7938, 8442, 8961, 9496, 10046, 10612, 11193, 11790, 12402, 13030, 13673, 14332
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #3.
- Reticular Chemistry Structure Resource (RCSR), The tcd tiling (or net)
- Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Crossrefs
See A299288 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.
Programs
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Mathematica
LinearRecurrence[{2, 0, -2, 1}, {1, 10, 33, 72, 126}, 50] (* Paolo Xausa, Aug 28 2024 *) -
PARI
Vec((1 + 8*x + 13*x^2 + 8*x^3 + x^4) / ((1 - x)^3*(1 + x)) + O(x^60)) \\ Colin Barker, Feb 11 2018
Formula
G.f.: (x^4 + 8*x^3 + 13*x^2 + 8*x + 1) / ((1 + x)*(1 - x)^3).
From Colin Barker, Feb 11 2018: (Start)
a(n) = (31*n^2 + 8) / 4 for even n>0.
a(n) = (31*n^2 + 9) / 4 for odd n>0.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>4. (End)
E.g.f.: ((8 + 31*x + 31*x^2)*cosh(x) + (9 + 31*x + 31*x^2)*sinh(x) - 4)/4. - Stefano Spezia, Jun 08 2024
Comments