A299258 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.6.12 2D tiling (cf. A072154).

1, 5, 13, 25, 41, 62, 89, 121, 157, 197, 242, 293, 349, 409, 473, 542, 617, 697, 781, 869, 962, 1061, 1165, 1273, 1385, 1502, 1625, 1753, 1885, 2021, 2162, 2309, 2461, 2617, 2777, 2942, 3113, 3289, 3469, 3653, 3842, 4037, 4237, 4441, 4649, 4862, 5081, 5305, 5533, 5765, 6002, 6245, 6493, 6745

Offset

0,2

References

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #23.

Links

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

Reticular Chemistry Structure Resource (RCSR), The fst 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^2-x+1)*(x+1)^3 / ((x^4+x^3+x^2+x+1)*(1-x)^3).

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 09 2018

a(n) ~ 12*n^2/5. - Stefano Spezia, Jun 06 2024

Mathematica

LinearRecurrence[{2, -1, 0, 0, 1, -2, 1}, {1, 5, 13, 25, 41, 62, 89, 121}, 60] (* Harvey P. Dale, Mar 14 2023 *)

Prog

(PARI) Vec((1 + x)^3*(1 - x + x^2)*(1 + x + x^2) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Feb 09 2018

Crossrefs

Cf. A072154.

Partial sums: A299264.

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: A099776 A301302 A133322 * A301671 A373059 A268525

Adjacent sequences: A299255 A299256 A299257 * A299259 A299260 A299261

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 07 2018

Status

approved