A299259 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.8.8 2D tiling (cf. A008576).

1, 5, 13, 26, 45, 69, 98, 133, 173, 218, 269, 325, 386, 453, 525, 602, 685, 773, 866, 965, 1069, 1178, 1293, 1413, 1538, 1669, 1805, 1946, 2093, 2245, 2402, 2565, 2733, 2906, 3085, 3269, 3458, 3653, 3853, 4058, 4269, 4485, 4706, 4933, 5165, 5402, 5645, 5893, 6146, 6405, 6669, 6938, 7213, 7493

Offset: 0

Author

N. J. A. Sloane, Feb 07 2018

Keywords

• nonn
• easy

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 #24.
• Reticular Chemistry Structure Resource (RCSR), The fee tiling (or net)
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Crossrefs

Cf. A008576, A061347.

Partial sums give A299265.

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

Magma

I:=[13, 26, 45, 69, 98]; [1,5] cat [n le 5 select I[n] else 2*Self(n-1) - Self(n-2) + Self(n-3) - 2*Self(n-4) + Self(n-5): n in [1..30]]; // G. C. Greubel, Feb 20 2018

Mathematica

CoefficientList[Series[(x+1)^3*(x^2+1)/((1-x)^3*(x^2+x+1)), {x, 0, 50}], x] (* G. C. Greubel, Feb 20 2018 *)

PARI

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

Formula

G.f.: (x + 1)^3*(x^2 + 1) / ((1 - x)^3*(x^2 + x + 1)).

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>5. - Colin Barker, Feb 09 2018

a(n) = (4*(5 + 6*n^2) + A061347(n))/9 for n > 0. - Stefano Spezia, Feb 17 2024