OFFSET
0,2
COMMENTS
Euler transform of length 4 sequence [6, -2, 1, -1]. - Michael Somos, Oct 03 2018
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).
FORMULA
From Colin Barker, Feb 09 2018: (Start)
G.f.: (1 + x)^3*(1 + x^2) / ((1 - x)^4*(1 + x + x^2)).
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6) for n>5.
(End)
a(n) = -a(-1-n) for all n in Z. - Michael Somos, Oct 03 2018
MATHEMATICA
CoefficientList[Series[(1+x)^3*(1+x^2)/((1-x)^4*(1+x+x^2)), {x, 0, 50}], x] (* G. C. Greubel, Feb 20 2018 *)
a[ n_] := (8 n^3 + 12 n^2 + 24 n + 9 + Mod[n, 3]) / 9; (* Michael Somos, Oct 03 2018 *)
LinearRecurrence[{3, -3, 2, -3, 3, -1}, {1, 6, 19, 45, 90, 159}, 50] (* Harvey P. Dale, Dec 11 2018 *)
PROG
(PARI) Vec((1 + x)^3*(1 + x^2) / ((1 - x)^4*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, Feb 09 2018
(PARI) {a(n) = (8*n^3 + 12*n^2 + 24*n + 9 + (n%3)) / 9}; /* Michael Somos, Oct 03 2018 */
(Magma) I:=[19, 45, 90, 159, 257, 390]; [1, 6] cat [n le 6 select I[n] else 3*Self(n-1) - 3*Self(n-2) +2*Self(n-3) - 3*Self(n-4) + 3*Self(n-5) - Self(n-6): n in [1..30]];
CROSSREFS
Cf. A299259.
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