A299285 Coordination sequence for "tea" 3D uniform tiling.

1, 10, 33, 73, 128, 199, 285, 388, 506, 640, 789, 955, 1136, 1333, 1545, 1774, 2018, 2278, 2553, 2845, 3152, 3475, 3813, 4168, 4538, 4924, 5325, 5743, 6176, 6625, 7089, 7570, 8066, 8578, 9105, 9649, 10208, 10783, 11373, 11980, 12602, 13240, 13893

Offset: 0

Author

N. J. A. Sloane, Feb 10 2018

Keywords

• nonn
• easy

Comments

First 20 terms computed by Davide M. Proserpio using ToposPro.

Links

• B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #4.
• Reticular Chemistry Structure Resource (RCSR), The tea tiling (or net)
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Crossrefs

See A299286 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.

Cf. A056594.

Programs

Mathematica

LinearRecurrence[{2,-1,0,1,-2,1},{1,10,33,73,128,199,285},50] (* Harvey P. Dale, May 09 2022 *)

PARI

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; 1,-2,1,0,-1,2]^n*[1;10;33;73;128;199])[1,1] \\ Charles R Greathouse IV, Oct 18 2022

Formula

From Colin Barker, Feb 11 2018: (Start)

G.f.: (1 + 8*x + 14*x^2 + 17*x^3 + 14*x^4 + 8*x^5 + x^6) / ((1 - x)^3*(1 + x)*(1 + x^2)).

a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>6. (End)

[I suspect Barker's formulas only conjectures. - N. J. A. Sloane, Jun 12 2024]

If the above formulas are true, then a(n) = (31 - 3*(-1)^n + 126*n^2 + 4*A056594(n))/16 for n > 0. - Stefano Spezia, Jun 08 2024