A001845 -id:A001845 - cp's OEIS frontend

A299275 Partial sums of A299274.

1, 5, 14, 32, 62, 109, 178, 269, 394, 554, 745, 983, 1265, 1596, 1987, 2435, 2943, 3525, 4175, 4884, 5674, 6551, 7515, 8562, 9702, 10955, 12308, 13771, 15331, 16998, 18799, 20707, 22750, 24915, 27212, 29683, 32263, 35000, 37893, 40913, 44115, 47459, 50988, 54674, 58530, 62612, 66817

Offset: 0

N. J. A. Sloane, Feb 10 2018

nonn

First 120 terms computed by Davide M. Proserpio using ToposPro.

V. A. Blatov, A. P. Shevchenko, D. M. Proserpio, Applied Topological Analysis of Crystal Structures with the Program Package ToposPro, Cryst. Growth Des. 2014, 14, 3576-3586.
Reticular Chemistry Structure Resource (RCSR), The hal tiling (or net)

Cf. A299274.

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.

There is a conjectured g.f., see the g.f. for A299274 and divide by 1-x. Note: this should not be used to generate a b-file. - N. J. A. Sloane, Feb 13 2018

A299276 Partial sums of A008137.

Original entry on oeis.org

1, 5, 14, 31, 59, 101, 161, 242, 347, 479, 641, 837, 1070, 1343, 1659, 2021, 2433, 2898, 3419, 3999, 4641, 5349, 6126, 6975, 7899, 8901, 9985, 11154, 12411, 13759, 15201, 16741, 18382, 20127, 21979, 23941, 26017, 28210, 30523, 32959, 35521, 38213, 41038

Offset: 0

N. J. A. Sloane, Feb 10 2018

Euler transform of length 6 sequence [5, -1, 1, -1, 1, -1]. - Michael Somos, Oct 03 2018

			G.f. = 1 + 5*x + 14*x^2 + 31*x^3 + 59*x^4 + 101*x^5 + 161*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. A008137.

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 n^3 + 12 n^2 + 40 n + 18 - {3, 3, 0, -3, -3, 3}[[Mod[n, 5] + 1]]) / 15; (* Michael Somos, Oct 03 2018 *)

Vec((1 + x)^3*(1 - x + x^2)*(1 + x^2) / ((1 - x)^4*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Feb 11 2018

{a(n) = (8*n^3 + 12*n^2 + 40*n + 18 - 3*(n%5<2) + 3*(n%5>2)) / 15}; /* Michael Somos, Oct 03 2018 */

From Colin Barker, Feb 11 2018: (Start)
G.f.: (1 + x)^3*(1 - x + x^2)*(1 + 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.

A299277 Coordination sequence for "pcu-i" 3D uniform tiling.

Original entry on oeis.org

1, 5, 13, 26, 46, 73, 104, 140, 187, 240, 292, 352, 417, 482, 567, 660, 740, 838, 944, 1031, 1150, 1290, 1399, 1531, 1677, 1787, 1944, 2130, 2261, 2431, 2624, 2750, 2941, 3180, 3334, 3538, 3777, 3920, 4149, 4440, 4610, 4852, 5144, 5297, 5560, 5910, 6097, 6373, 6717, 6881, 7182, 7590, 7787, 8101, 8504, 8672

Offset: 0

N. J. A. Sloane, Feb 10 2018

First 80 terms computed by Davide M. Proserpio using ToposPro.

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #20.

Colin Barker, Table of n, a(n) for n = 0..1000
V. A. Blatov, A. P. Shevchenko, D. M. Proserpio, Applied Topological Analysis of Crystal Structures with the Program Package ToposPro, Cryst. Growth Des. 2014, 14, 3576-3586.
Reticular Chemistry Structure Resource (RCSR), The pcu-i tiling (or net)
Index entries for linear recurrences with constant coefficients, signature (0,-1,1,0,1,2,0,2,-2,0,-2,-1,0,-1,1,0,1).

See A299278 for partial sums.

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.

CoefficientList[Series[(x^16-x^15+x^14-2x^13+2x^12-x^11+4x^10+x^9+9x^8+12x^6-x^5+ 9x^4+4x^2+1)(x+1)^5/((1+x^2)(1-x^3)(1-x^6)^2),{x,0,60}],x] (* or *) LinearRecurrence[{ 2,-4,7,-10,14,-16,18,-18,16,-14,10,-7,4,-2,1},{1,5,13,26,46,73,104,140,187,240,292,352,417,482,567,660,740,838,944,1031},60] (* Harvey P. Dale, Mar 09 2024 *)

Vec((x^16 - x^15 + x^14 - 2*x^13 + 2*x^12 - x^11 + 4*x^10 + x^9 + 9*x^8 + 12*x^6 - x^5 + 9*x^4 + 4*x^2 + 1) * (x + 1)^5 / ((1 + x^2)*(1 - x^3)*(1 - x^6)^2) + O(x^60)) \\ Colin Barker, Feb 14 2018

G.f.: (x^16 - x^15 + x^14 - 2*x^13 + 2*x^12 - x^11 + 4*x^10 + x^9 + 9*x^8 + 12*x^6 - x^5 + 9*x^4 + 4*x^2 + 1) * (x + 1)^5 / ((1 + x^2)*(1 - x^3)*(1 - x^6)^2). - N. J. A. Sloane, Feb 13 2018

a(n) = -a(n-2) + a(n-3) + a(n-5) + 2*a(n-6) + 2*a(n-8) - 2*a(n-9) - 2*a(n-11) - a(n-12) - a(n-14) + a(n-15) + a(n-17) for n>21. - Colin Barker, Feb 14 2018

A299278 Partial sums of A299277.

Original entry on oeis.org

1, 6, 19, 45, 91, 164, 268, 408, 595, 835, 1127, 1479, 1896, 2378, 2945, 3605, 4345, 5183, 6127, 7158, 8308, 9598, 10997, 12528, 14205, 15992, 17936, 20066, 22327, 24758, 27382, 30132, 33073, 36253, 39587, 43125, 46902, 50822, 54971, 59411, 64021, 68873, 74017, 79314, 84874

Offset: 0

N. J. A. Sloane, Feb 10 2018

First 80 terms computed by Davide M. Proserpio using ToposPro.

Colin Barker, Table of n, a(n) for n = 0..1000
V. A. Blatov, A. P. Shevchenko, D. M. Proserpio, Applied Topological Analysis of Crystal Structures with the Program Package ToposPro, Cryst. Growth Des. 2014, 14, 3576-3586.
Reticular Chemistry Structure Resource (RCSR), The pcu-i tiling (or net)
Index entries for linear recurrences with constant coefficients, signature (1,-1,2,-1,1,1,-2,2,-4,2,-2,1,1,-1,2,-1,1,-1).

Cf. A299277.

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((x^16 - x^15 + x^14 - 2*x^13 + 2*x^12 - x^11 + 4*x^10 + x^9 + 9*x^8 + 12*x^6 - x^5 + 9*x^4 + 4*x^2 + 1) * (x + 1)^5 / ((1 - x)*(1 + x^2)*(1 - x^3)*(1 - x^6)^2) + O(x^60)) \\ Colin Barker, Feb 14 2018

G.f.: (x^16 - x^15 + x^14 - 2*x^13 + 2*x^12 - x^11 + 4*x^10 + x^9 + 9*x^8 + 12*x^6 - x^5 + 9*x^4 + 4*x^2 + 1) * (x + 1)^5 / ((1 - x)*(1 + x^2)*(1 - x^3)*(1 - x^6)^2). - N. J. A. Sloane, Feb 13 2018

a(n) = a(n-1) - a(n-2) + 2*a(n-3) - a(n-4) + a(n-5) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 4*a(n-9) + 2*a(n-10) - 2*a(n-11) + a(n-12) + a(n-13) - a(n-14) + 2*a(n-15) - a(n-16) + a(n-17) - a(n-18) for n>21. - Colin Barker, Feb 14 2018

A299280 Partial sums of A299279.

Original entry on oeis.org

1, 9, 39, 107, 233, 413, 699, 1047, 1557, 2129, 2927, 3779, 4929, 6117, 7683, 9263, 11309, 13337, 15927, 18459, 21657, 24749, 28619, 32327, 36933, 41313, 46719, 51827, 58097, 63989, 71187, 77919, 86109, 93737, 102983, 111563, 121929, 131517, 143067, 153719, 166517

Offset: 0

N. J. A. Sloane, Feb 10 2018

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

Cf. A299279.

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[{1,3,-3,-3,3,1,-1},{1,9,39,107,233,413,699,1047},50] (* Harvey P. Dale, Jul 22 2021 *)

Vec((1 + 8*x + 27*x^2 + 44*x^3 + 39*x^4 - 3*x^6 + 4*x^7) / ((1 - x)^4*(1 + x)^3) + O(x^60)) \\ Colin Barker, Feb 11 2018

From Colin Barker, Feb 11 2018: (Start)
G.f.: (1 + 8*x + 27*x^2 + 44*x^3 + 39*x^4 - 3*x^6 + 4*x^7) / ((1 - x)^4*(1 + x)^3).
a(n) = (5*n^3 + 8*n^2 + 6*n - 6) / 2 for n>0 and even.
a(n) = (5*n^3 + 7*n^2 + 5*n + 1) / 2 for n odd.
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>7. (End)
E.g.f.: (8 - (6 - 17*x - 23*x^2 - 5*x^3)*cosh(x) + (1 + 19*x + 22*x^2 + 5*x^3)*sinh(x))/2. - Stefano Spezia, Jun 06 2024

A299281 Coordination sequence for "reo-e" 3D uniform tiling.

Original entry on oeis.org

1, 6, 19, 41, 72, 114, 166, 224, 288, 364, 454, 550, 648, 758, 886, 1020, 1152, 1296, 1462, 1634, 1800, 1978, 2182, 2392, 2592, 2804, 3046, 3294, 3528, 3774, 4054, 4340, 4608, 4888, 5206, 5530, 5832, 6146, 6502, 6864, 7200, 7548, 7942, 8342, 8712, 9094, 9526, 9964, 10368

Offset: 0

N. J. A. Sloane, Feb 10 2018

First 20 terms computed by Davide M. Proserpio using ToposPro.

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #9.

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

See A299282 for partial sums.
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.
Cf. A056594.

Vec((1 + x)*(1 + x^2 + x^3)*(1 + 2*x + 3*x^2 + x^4 - 2*x^5 + x^6) / ((1 - x)^3*(1 + x^2)^2) + O(x^60)) \\ Colin Barker, Feb 14 2018

G.f.: (x+1)*(x^3+x^2+1)*(x^6-2*x^5+x^4+3*x^2+2*x+1) / ((x^2+1)^2*(1-x)^3). - N. J. A. Sloane, Feb 12 2018
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>8. - Colin Barker, Feb 14 2018
a(n) = (9*n^2 + 4*(1 - A056594(n)) - (n - 4)*A056594(n+1))/2 for n > 3. - Stefano Spezia, Apr 23 2023

a(21)-a(40) from Davide M. Proserpio, Feb 12 2018

A299282 Partial sums of A299281.

Original entry on oeis.org

1, 7, 26, 67, 139, 253, 419, 643, 931, 1295, 1749, 2299, 2947, 3705, 4591, 5611, 6763, 8059, 9521, 11155, 12955, 14933, 17115, 19507, 22099, 24903, 27949, 31243, 34771, 38545, 42599, 46939, 51547, 56435, 61641, 67171, 73003, 79149, 85651, 92515, 99715, 107263, 115205

Offset: 0

N. J. A. Sloane, Feb 10 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. A056594, A299281.

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^2 + x^3)*(1 + 2*x + 3*x^2 + x^4 - 2*x^5 + x^6) / ((1 - x)^4*(1 + x^2)^2) + O(x^70)) \\ Colin Barker, Feb 14 2018

From Colin Barker, Feb 14 2018: (Start)
G.f.: (1 + x)*(1 + x^2 + x^3)*(1 + 2*x + 3*x^2 + x^4 - 2*x^5 + x^6) / ((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)
a(n) = (n*(6*n^2 + 9*n + 11) - 12 + (n - 8)*A056594(n) - (n + 1)*A056594(n+1))/4 for n > 2. - Stefano Spezia, Apr 23 2023

A299283 Coordination sequence for "svh" 3D uniform tiling.

Original entry on oeis.org

1, 7, 22, 48, 84, 130, 186, 253, 330, 417, 514, 622, 740, 868, 1006, 1155, 1314, 1483, 1662, 1852, 2052, 2262, 2482, 2713, 2954, 3205, 3466, 3738, 4020, 4312, 4614, 4927, 5250, 5583, 5926, 6280, 6644, 7018, 7402, 7797, 8202, 8617, 9042, 9478, 9924, 10380

Offset: 0

N. J. A. Sloane, Feb 10 2018

First 20 terms computed by Davide M. Proserpio using ToposPro.

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #15.

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

See A299284 for partial sums.

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,-1,0,1,-2,1},{1,7,22,48,84,130,186},50] (* Harvey P. Dale, May 19 2019 *)

Vec((1 + 5*x + 9*x^2 + 11*x^3 + 9*x^4 + 5*x^5 + x^6) / ((1 - x)^3*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Feb 11 2018

G.f.: (x^6+5*x^5+9*x^4+11*x^3+9*x^2+5*x+1)/((x+1)*(x^2+1)*(1-x)^3).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>6. - Colin Barker, Feb 11 2018
a(n) = (29 - (-1)^n + 82*n^2 + 4*A056594(n))/16 for n > 0. - Stefano Spezia, Jun 06 2024

A299284 Partial sums of A299283.

Original entry on oeis.org

1, 8, 30, 78, 162, 292, 478, 731, 1061, 1478, 1992, 2614, 3354, 4222, 5228, 6383, 7697, 9180, 10842, 12694, 14746, 17008, 19490, 22203, 25157, 28362, 31828, 35566, 39586, 43898, 48512, 53439, 58689, 64272, 70198, 76478, 83122, 90140, 97542, 105339, 113541

Offset: 0

N. J. A. Sloane, Feb 10 2018

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

Cf. A299283.

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,-3,1,1,-3,3,-1},{1,8,30,78,162,292,478},50] (* Harvey P. Dale, Mar 30 2024 *)

Vec((1 + 5*x + 9*x^2 + 11*x^3 + 9*x^4 + 5*x^5 + x^6) / ((1 - x)^4*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Feb 11 2018

From Colin Barker, Feb 11 2018: (Start)
G.f.: (1 + 5*x + 9*x^2 + 11*x^3 + 9*x^4 + 5*x^5 + x^6) / ((1 - x)^4*(1 + x)*(1 + x^2)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-4) - 3*a(n-5) + 3*a(n-6) - a(n-7) for n>6.
(End)

A299285 Coordination sequence for "tea" 3D uniform tiling.

Original entry on oeis.org

1, 10, 33, 73, 128, 199, 285, 388, 506, 640, 789, 955, 1136, 1333, 1545, 1774, 2018, 2278, 2553, 2845, 3152, 3475, 3813, 4168, 4538, 4924, 5325, 5743, 6176, 6625, 7089, 7570, 8066, 8578, 9105, 9649, 10208, 10783, 11373, 11980, 12602, 13240, 13893

Offset: 0

N. J. A. Sloane, Feb 10 2018

First 20 terms computed by Davide M. Proserpio using ToposPro.

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

See A299286 for partial sums.
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.
Cf. A056594.

LinearRecurrence[{2,-1,0,1,-2,1},{1,10,33,73,128,199,285},50] (* Harvey P. Dale, May 09 2022 *)

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; 1,-2,1,0,-1,2]^n*[1;10;33;73;128;199])[1,1] \\ Charles R Greathouse IV, Oct 18 2022

From Colin Barker, Feb 11 2018: (Start)
G.f.: (1 + 8*x + 14*x^2 + 17*x^3 + 14*x^4 + 8*x^5 + x^6) / ((1 - x)^3*(1 + x)*(1 + x^2)).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>6. (End)
[I suspect Barker's formulas only conjectures. - N. J. A. Sloane, Jun 12 2024]
If the above formulas are true, then a(n) = (31 - 3*(-1)^n + 126*n^2 + 4*A056594(n))/16 for n > 0. - Stefano Spezia, Jun 08 2024

cp's OEIS Frontend

A299275 Partial sums of A299274.

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A299276 Partial sums of A008137.

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A299277 Coordination sequence for "pcu-i" 3D uniform tiling.

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A299278 Partial sums of A299277.

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A299280 Partial sums of A299279.

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A299281 Coordination sequence for "reo-e" 3D uniform tiling.

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A299282 Partial sums of A299281.

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A299283 Coordination sequence for "svh" 3D uniform tiling.

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