A299256 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.6.3.6 2D tiling (cf. A008579).

1, 6, 18, 40, 72, 112, 162, 220, 288, 364, 450, 544, 648, 760, 882, 1012, 1152, 1300, 1458, 1624, 1800, 1984, 2178, 2380, 2592, 2812, 3042, 3280, 3528, 3784, 4050, 4324, 4608, 4900, 5202, 5512, 5832, 6160, 6498, 6844, 7200, 7564, 7938, 8320, 8712, 9112, 9522, 9940, 10368, 10804, 11250, 11704

Offset: 0

Author

N. J. A. Sloane, Feb 07 2018

Keywords

• nonn
• easy

References

• B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #18.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Reticular Chemistry Structure Resource (RCSR), The kag tiling (or net)
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A008579.

For partial sums see A299262.

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

GAP

a:=[18,40,72,112];; for n in [5..50] do a[n]:=2*a[n-1]-2*a[n-3]+a[n-4]; od; Concatenation([1,6],a); # Muniru A Asiru, Oct 26 2018

Magma

[1, 6] cat [9*n^2 div 2: n in [2..50]]; // Vincenzo Librandi, Oct 26 2018

Maple

seq(coeff(series((1+2*x)*(x^4-2*x^3-2*x^2-2*x-1)/((x-1)^3*(1+x)),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 26 2018

Mathematica

Join[{1, 6}, LinearRecurrence[{2, 0, -2, 1}, {18, 40, 72, 112}, 50]] (* Vincenzo Librandi, Oct 26 2018 *)

PARI

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

Formula

G.f.: (1 + 2*x)*(x^4 - 2*x^3 - 2*x^2 - 2*x - 1) / ((x - 1)^3*(x + 1)).

From Colin Barker, Feb 09 2018: (Start)

a(n) = 9*n^2 / 2 for n>1.

a(n) = (9*n^2 - 1) / 2 for n>1.

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>5. (End)

E.g.f.: (2 + 4*x + 9*x*(x + 1)*cosh(x) + (9*x^2 + 9*x - 1)*sinh(x))/2. - Stefano Spezia, Mar 14 2024