A299254 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3^4.6 2D tiling (cf. A250120).

1, 7, 21, 45, 79, 122, 175, 237, 309, 391, 482, 583, 693, 813, 943, 1082, 1231, 1389, 1557, 1735, 1922, 2119, 2325, 2541, 2767, 3002, 3247, 3501, 3765, 4039, 4322, 4615, 4917, 5229, 5551, 5882, 6223, 6573, 6933, 7303, 7682, 8071, 8469, 8877, 9295, 9722, 10159, 10605, 11061, 11527, 12002

Offset

0,2

References

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #17.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Reticular Chemistry Structure Resource (RCSR), The svj tiling (or net)

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Formula

G.f.: (x^2+x+1)*(x^4+3*x^3+3*x+1)*(x+1) / ((x^4+x^3+x^2+x+1)*(1-x)^3). (This is the product of the g.f.'s for A250120 and A040000. - N. J. A. Sloane, Nov 10 2018)

a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) for n>7. - Colin Barker, Feb 07 2018

a(n) = 2*((sqrt(5) - 5)*(5 + 12*n^2) - (sqrt(5) - 1)*cos(2*n*Pi/5) + (sqrt(5) - 1)*cos(4*n*Pi/5))/(5*(sqrt(5) - 5)) for n > 0. - Stefano Spezia, Jun 06 2024

Mathematica

LinearRecurrence[{2, -1, 0, 0, 1, -2, 1}, {1, 7, 21, 45, 79, 122, 175, 237}, 50] (* Paolo Xausa, Jan 16 2025 *)

Prog

(PARI) Vec((1 + x)*(1 + x + x^2)*(1 + 3*x + 3*x^3 + x^4) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Feb 07 2018

Crossrefs

Cf. A040000, A250120.

Partial sums: A299260.

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: A387625 A256051 A388000 * A146411 A287431 A127736

Adjacent sequences: A299251 A299252 A299253 * A299255 A299256 A299257

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 06 2018

Status

approved