A299265 Partial sums of A299259.

1, 6, 19, 45, 90, 159, 257, 390, 563, 781, 1050, 1375, 1761, 2214, 2739, 3341, 4026, 4799, 5665, 6630, 7699, 8877, 10170, 11583, 13121, 14790, 16595, 18541, 20634, 22879, 25281, 27846, 30579, 33485, 36570, 39839, 43297, 46950, 50803, 54861, 59130

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.

Sequence in context: A272707 A266938 A362602 * A005712 A299278 A298741

Adjacent sequences: A299262 A299263 A299264 * A299266 A299267 A299268

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 07 2018

Status

approved