OFFSET
0,2
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).
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
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 *)
PROG
(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
(Magma) [1, 6] cat [9*n^2 div 2: n in [2..50]]; // Vincenzo Librandi, Oct 26 2018
(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
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.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Feb 07 2018
STATUS
approved