A299261 Partial sums of A299255.

1, 8, 31, 81, 168, 303, 497, 760, 1103, 1537, 2072, 2719, 3489, 4392, 5439, 6641, 8008, 9551, 11281, 13208, 15343, 17697, 20280, 23103, 26177, 29512, 33119, 37009, 41192, 45679, 50481, 55608, 61071, 66881, 73048, 79583, 86497, 93800, 101503, 109617

Offset: 0

Author

N. J. A. Sloane, Feb 07 2018

Keywords

• nonn
• easy

Comments

Euler transform of length 3 sequence [8, -5, 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).

Crossrefs

Cf. A299255.

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

Mathematica

a[ n_] := (8 (2 n + 1) (n^2 + n + 1) - Mod[n - 1, 3, -1]) / 9; (* Michael Somos, Oct 03 2018 *)
LinearRecurrence[{3,-3,2,-3,3,-1},{1,8,31,81,168,303},50] (* Harvey P. Dale, May 15 2025 *)

PARI

Vec((1 + x)^5 / ((1 - x)^4*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, Feb 09 2018

PARI

{a(n) =  (8 * (2*n + 1) * (n^2 + n + 1) + (n%3==0) - (n%3==2)) / 9}; /* Michael Somos, Oct 03 2018 */

Formula

From Colin Barker, Feb 09 2018: (Start)

G.f.: (1 + x)^5 / ((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