A299260 Partial sums of A299254.

1, 8, 29, 74, 153, 275, 450, 687, 996, 1387, 1869, 2452, 3145, 3958, 4901, 5983, 7214, 8603, 10160, 11895, 13817, 15936, 18261, 20802, 23569, 26571, 29818, 33319, 37084, 41123, 45445, 50060, 54977, 60206, 65757, 71639, 77862, 84435, 91368, 98671, 106353, 114424

Offset: 0

Author

N. J. A. Sloane, Feb 07 2018

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,1,-3,3,-1).

Crossrefs

Cf. A299254.

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

LinearRecurrence[{3, -3, 1, 0, 1, -3, 3, -1}, {1, 8, 29, 74, 153, 275, 450, 687}, 50] (* Paolo Xausa, Jan 16 2025 *)

PARI

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

Formula

From Colin Barker, Feb 09 2018: (Start)

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

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5) - 3*a(n-6) + 3*a(n-7) - a(n-8) for n>7.

(End)

a(n) = (1/5)*(8*n^3 + 12*n^2 + 14*n + 5 + [n == 1 (mod 5)] - [n == 3 (mod 5)]). - Eric Simon Jacob, Feb 14 2023