A005897 -id:A005897 - cp's OEIS frontend

A299256 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.6.3.6 2D tiling (cf. A008579).

1, 6, 18, 40, 72, 112, 162, 220, 288, 364, 450, 544, 648, 760, 882, 1012, 1152, 1300, 1458, 1624, 1800, 1984, 2178, 2380, 2592, 2812, 3042, 3280, 3528, 3784, 4050, 4324, 4608, 4900, 5202, 5512, 5832, 6160, 6498, 6844, 7200, 7564, 7938, 8320, 8712, 9112, 9522, 9940, 10368, 10804, 11250, 11704

Offset: 0

N. J. A. Sloane, Feb 07 2018

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #18.

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).

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.

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

[1, 6] cat [9*n^2 div 2: n in [2..50]]; // Vincenzo Librandi, Oct 26 2018

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

Join[{1, 6}, LinearRecurrence[{2, 0, -2, 1}, {18, 40, 72, 112}, 50]] (* Vincenzo Librandi, Oct 26 2018 *)

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

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

A299257 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.12.12 2D tiling (cf. A250122).

Original entry on oeis.org

1, 5, 12, 22, 36, 56, 82, 111, 144, 183, 226, 272, 324, 382, 442, 505, 576, 653, 730, 810, 900, 996, 1090, 1187, 1296, 1411, 1522, 1636, 1764, 1898, 2026, 2157, 2304, 2457, 2602, 2750, 2916, 3088, 3250, 3415, 3600, 3791, 3970, 4152, 4356, 4566, 4762, 4961, 5184, 5413, 5626, 5842, 6084, 6332

Offset: 0

N. J. A. Sloane, Feb 07 2018

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #19.

Colin Barker, Table of n, a(n) for n = 0..1000
Reticular Chemistry Structure Resource (RCSR), The ttw tiling (or net)
Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Cf. A250122.
Partial sums: A299263.
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.

LinearRecurrence[{3, -5, 7, -7, 5, -3, 1}, {1, 5, 12, 22, 36, 56, 82, 111, 144, 183}, 60] (* Paolo Xausa, Jun 20 2024 *)

Vec((1 + x)*(1 + x + x^2 + 3*x^3 - x^4 + 5*x^5 - 3*x^6 + 4*x^7 - 2*x^8) / ((1 - x)^3*(1 + x^2)^2) + O(x^60)) \\ Colin Barker, Feb 09 2018

G.f.: (2*x^8 - 4*x^7 + 3*x^6 - 5*x^5 + x^4 - 3*x^3 - x^2 - x - 1)*(x + 1) / ((x - 1)^3*(x^2 + 1)^2).
From Colin Barker, Feb 09 2018: (Start)
a(n) = (4 - (2+8*i)*(-i)^n - (2-8*i)*i^n + i*((-i)^n-i^n)*n + 18*n^2) / 8 for n>2, where i=sqrt(-1).
a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7) for n>9. (End)
a(n) = 1/2 + 9*n^2/4 + (-1)^floor(n/2)*(A027656(n-1)/2 - A010699(n)/4). - R. J. Mathar, Feb 12 2021

A299258 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.6.12 2D tiling (cf. A072154).

Original entry on oeis.org

1, 5, 13, 25, 41, 62, 89, 121, 157, 197, 242, 293, 349, 409, 473, 542, 617, 697, 781, 869, 962, 1061, 1165, 1273, 1385, 1502, 1625, 1753, 1885, 2021, 2162, 2309, 2461, 2617, 2777, 2942, 3113, 3289, 3469, 3653, 3842, 4037, 4237, 4441, 4649, 4862, 5081, 5305, 5533, 5765, 6002, 6245, 6493, 6745

Offset: 0

N. J. A. Sloane, Feb 07 2018

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #23.

Colin Barker, Table of n, a(n) for n = 0..1000
Reticular Chemistry Structure Resource (RCSR), The fst tiling (or net)
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A072154.
Partial sums: A299264.
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.

LinearRecurrence[{2,-1,0,0,1,-2,1},{1,5,13,25,41,62,89,121},60] (* Harvey P. Dale, Mar 14 2023 *)

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

G.f.: (x^2+x+1)*(x^2-x+1)*(x+1)^3 / ((x^4+x^3+x^2+x+1)*(1-x)^3).
a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) for n>7. - Colin Barker, Feb 09 2018
a(n) ~ 12*n^2/5. - Stefano Spezia, Jun 06 2024

A299259 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.8.8 2D tiling (cf. A008576).

Original entry on oeis.org

1, 5, 13, 26, 45, 69, 98, 133, 173, 218, 269, 325, 386, 453, 525, 602, 685, 773, 866, 965, 1069, 1178, 1293, 1413, 1538, 1669, 1805, 1946, 2093, 2245, 2402, 2565, 2733, 2906, 3085, 3269, 3458, 3653, 3853, 4058, 4269, 4485, 4706, 4933, 5165, 5402, 5645, 5893, 6146, 6405, 6669, 6938, 7213, 7493

Offset: 0

N. J. A. Sloane, Feb 07 2018

Colin Barker, Table of n, a(n) for n = 0..1000
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #24.
Reticular Chemistry Structure Resource (RCSR), The fee tiling (or net)
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A008576, A061347.
Partial sums give A299265.
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.

I:=[13, 26, 45, 69, 98]; [1,5] cat [n le 5 select I[n] else 2*Self(n-1) - Self(n-2) + Self(n-3) - 2*Self(n-4) + Self(n-5): n in [1..30]]; // G. C. Greubel, Feb 20 2018

CoefficientList[Series[(x+1)^3*(x^2+1)/((1-x)^3*(x^2+x+1)), {x, 0, 50}], x] (* G. C. Greubel, Feb 20 2018 *)

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

G.f.: (x + 1)^3*(x^2 + 1) / ((1 - x)^3*(x^2 + x + 1)).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>5. - Colin Barker, Feb 09 2018
a(n) = (4*(5 + 6*n^2) + A061347(n))/9 for n > 0. - Stefano Spezia, Feb 17 2024

A299260 Partial sums of A299254.

Original entry on oeis.org

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

N. J. A. Sloane, Feb 07 2018

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).

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.

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

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

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

A299261 Partial sums of A299255.

Original entry on oeis.org

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

N. J. A. Sloane, Feb 07 2018

Euler transform of length 3 sequence [8, -5, 1]. - Michael Somos, Oct 03 2018

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).

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.

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 *)

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

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

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

A299262 Partial sums of A299256.

Original entry on oeis.org

1, 7, 25, 65, 137, 249, 411, 631, 919, 1283, 1733, 2277, 2925, 3685, 4567, 5579, 6731, 8031, 9489, 11113, 12913, 14897, 17075, 19455, 22047, 24859, 27901, 31181, 34709, 38493, 42543, 46867, 51475, 56375, 61577, 67089, 72921, 79081, 85579, 92423, 99623, 107187, 115125, 123445, 132157, 141269, 150791, 160731, 171099

Offset: 0

N. J. A. Sloane, Feb 07 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A299256.

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.

LinearRecurrence[{3,-2,-2,3,-1},{1,7,25,65,137,249},50] (* Harvey P. Dale, Jul 22 2024 *)

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

From Colin Barker, Feb 09 2018: (Start)
G.f.: (1 + 2*x)*(1 + 2*x + 2*x^2 + 2*x^3 - x^4) / ((1 - x)^4*(1 + x)).
a(n) = (6*n^3 + 9*n^2 + 2*n + 12) / 4 for n>0 and even.
a(n) = (6*n^3 + 9*n^2 + 2*n + 11) / 4 for n odd.
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) for n>5. (End)
E.g.f.: ((12 + 17*x + 27*x^2 + 6*x^3)*cosh(x) + (11 + 17*x + 27*x^2 + 6*x^3)*sinh(x) - 8)/4. - Stefano Spezia, Mar 14 2024

A299263 Partial sums of A299257.

Original entry on oeis.org

1, 6, 18, 40, 76, 132, 214, 325, 469, 652, 878, 1150, 1474, 1856, 2298, 2803, 3379, 4032, 4762, 5572, 6472, 7468, 8558, 9745, 11041, 12452, 13974, 15610, 17374, 19272, 21298, 23455, 25759, 28216, 30818, 33568, 36484, 39572, 42822, 46237, 49837, 53628, 57598

Offset: 0

N. J. A. Sloane, Feb 07 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-8,12,-14,12,-8,4,-1).

Cf. A299257.

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.

Vec((1 + x)*(1 + x + x^2 + 3*x^3 - x^4 + 5*x^5 - 3*x^6 + 4*x^7 - 2*x^8) / ((1 - x)^4*(1 + x^2)^2) + O(x^60)) \\ Colin Barker, Feb 09 2018

From Colin Barker, Feb 09 2018: (Start)
G.f.: (1 + x)*(1 + x + x^2 + 3*x^3 - x^4 + 5*x^5 - 3*x^6 + 4*x^7 - 2*x^8) / ((1 - x)^4*(1 + x^2)^2).
a(n) = 4*a(n-1) - 8*a(n-2) + 12*a(n-3) - 14*a(n-4) + 12*a(n-5) - 8*a(n-6) + 4*a(n-7) - a(n-8) for n>8.
(End)
5*a(n) = 2*(2*n+1)*(2*n^2+2*n+9)/3 - A138019(n). - R. J. Mathar, Feb 12 2021

A299264 Partial sums of A299258.

Original entry on oeis.org

1, 6, 19, 44, 85, 147, 236, 357, 514, 711, 953, 1246, 1595, 2004, 2477, 3019, 3636, 4333, 5114, 5983, 6945, 8006, 9171, 10444, 11829, 13331, 14956, 16709, 18594, 20615, 22777, 25086, 27547, 30164, 32941, 35883, 38996, 42285, 45754, 49407, 53249, 57286, 61523

Offset: 0

N. J. A. Sloane, Feb 07 2018

Euler transform of length 6 sequence [6, -2, 0, 0, 1, -1]. - Michael Somos, Oct 03 2018

			G.f. = 1 + 6*x + 19*x^2 + 44*x^3 + 85*x^4 + 147*x^5 + 236*x^6 + ... - _Michael Somos_, Oct 03 2018

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).

Cf. A299258.

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.

a[ n_] := (4 n^3 + 6 n^2 + 16 n + {5, 4, 7, 10, 9}[[Mod[n, 5] + 1]]) / 5; (* Michael Somos, Oct 03 2018 *)
LinearRecurrence[{3,-3,1,0,1,-3,3,-1},{1,6,19,44,85,147,236,357},50] (* Harvey P. Dale, Aug 03 2025 *)

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

{a(n) = (4*n^3 + 6*n^2 + 16*n + [5, 4, 7, 10, 9][n%5+1]) / 5}; /* Michael Somos, Oct 03 2018 */

From Colin Barker, Feb 09 2018: (Start)
G.f.: (1 + x)^3*(1 - x + x^2)*(1 + x + x^2) / ((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) = -a(-1-n) for all n in Z. - Michael Somos, Oct 03 2018
a(n) ~ 4*n^3/5. - Stefano Spezia, Jun 06 2024

A299265 Partial sums of A299259.

Original entry on oeis.org

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

N. J. A. Sloane, Feb 07 2018

Euler transform of length 4 sequence [6, -2, 1, -1]. - Michael Somos, Oct 03 2018

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).

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.

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]];

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 *)

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

{a(n) =  (8*n^3 + 12*n^2 + 24*n + 9 + (n%3)) / 9}; /* Michael Somos, Oct 03 2018 */

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

cp's OEIS Frontend

A299256 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.6.3.6 2D tiling (cf. A008579).

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A299257 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.12.12 2D tiling (cf. A250122).

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A299258 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.6.12 2D tiling (cf. A072154).

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A299259 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.8.8 2D tiling (cf. A008576).

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A299260 Partial sums of A299254.

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A299261 Partial sums of A299255.

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A299262 Partial sums of A299256.

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