A299256 -id:A299256 - cp's OEIS frontend

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

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

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

A299266 Coordination sequence for "cab" 3D uniform tiling formed from octahedra and truncated cubes.

Original entry on oeis.org

1, 5, 9, 22, 37, 57, 82, 117, 145, 178, 229, 281, 322, 377, 445, 514, 577, 645, 730, 825, 901, 982, 1093, 1205, 1294, 1397, 1525, 1654, 1765, 1881, 2026, 2181, 2305, 2434, 2605, 2777, 2914, 3065, 3253, 3442, 3601, 3765, 3970, 4185, 4357, 4534, 4765, 4997, 5182, 5381, 5629, 5878, 6085, 6297, 6562, 6837

Offset: 0

N. J. A. Sloane, Feb 07 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 #8.

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 cab tiling (or net)
Davide M. Proserpio, Summary of the 28 uniform 3D tilings and their coordination sequences (produced by ToposPro)
Index entries for linear recurrences with constant coefficients, signature (1,-1,2,0,0,0,-2,1,-1,1).

See A299267 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.

I:=[22, 37, 57, 82, 117, 145, 178,229, 281,322]; [1,5,9] cat [n le 10 select I[n] else Self(n-1) -Self(n-2) +2*Self(n-3)-2*Self(n-7)+Self(n-8)-Self(n-9) + Self(n-10): n in [1..30]]; // G. C. Greubel, Feb 20 2018

CoefficientList[Series[(4*x^12-4*x^11+x^10+5*x^8+20*x^7+18*x^6+24*x^5 +14*x^4+16*x^3+5*x^2+4*x+1)/((1-x)*(1-x^2)*(1-x^3)*(1+x^2)^2), {x,0, 50}], x] (* G. C. Greubel, Feb 20 2018 *)

Vec((1 + 4*x + 5*x^2 + 16*x^3 + 14*x^4 + 24*x^5 + 18*x^6 + 20*x^7 + 5*x^8 + x^10 - 4*x^11 + 4*x^12) / ((1 - x)^3*(1 + x)*(1 + x^2)^2*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, Feb 15 2018

G.f.: (4*x^12 -4*x^11 +x^10 +5*x^8 +20*x^7 +18*x^6 +24*x^5 +14*x^4 +16*x^3 +5*x^2 +4*x +1)/((1-x)*(1-x^2)*(1-x^3)*(1+x^2)^2). - N. J. A. Sloane, Feb 12 2018

a(n) = a(n-1) - a(n-2) + 2*a(n-3) - 2*a(n-7) + a(n-8) - a(n-9) + a(n-10) for n>12. - Colin Barker, Feb 15 2018

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

A299267 Partial sums of A299266.

Original entry on oeis.org

1, 6, 15, 37, 74, 131, 213, 330, 475, 653, 882, 1163, 1485, 1862, 2307, 2821, 3398, 4043, 4773, 5598, 6499, 7481, 8574, 9779, 11073, 12470, 13995, 15649, 17414, 19295, 21321, 23502, 25807, 28241, 30846, 33623, 36537, 39602, 42855, 46297, 49898, 53663, 57633, 61818, 66175, 70709, 75474, 80471, 85653, 91034, 96663

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 (2,-2,3,-2,0,0,-2,3,-2,2,-1).

Cf. A299266.

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:=[15,37,74,131,213,330,475,653,882,1163,1485]; [1,6] cat [n le 11 select I[n] else 2*Self(n-1) -2*Self(n-2) +3*Self(n-3)-2*Self(n-4)-2*Self(n-7) +3*Self(n-8) -2*Self(n-9)+2*Self(n-10)-Self(n-11): n in [1..30]]; // G. C. Greubel, Feb 20 2018

CoefficientList[Series[(1 +4*x +5*x^2 +16*x^3 +14*x^4 +24*x^5 +18*x^6 +20*x^7 +5*x^8 + x^10 -4*x^11 +4*x^12)/((1 -x)^4*(1 +x)*(1 +x^2)^2*(1 +x +x^2)), {x, 0, 50}], x] (* G. C. Greubel, Feb 20 2018 *)
LinearRecurrence[{2,-2,3,-2,0,0,-2,3,-2,2,-1},{1,6,15,37,74,131,213,330,475,653,882,1163,1485},60] (* Harvey P. Dale, Sep 03 2018 *)

Vec((1 + 4*x + 5*x^2 + 16*x^3 + 14*x^4 + 24*x^5 + 18*x^6 + 20*x^7 + 5*x^8 + x^10 - 4*x^11 + 4*x^12) / ((1 - x)^4*(1 + x)*(1 + x^2)^2*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, Feb 15 2018

From Colin Barker, Feb 15 2018: (Start)
G.f.: (1 +4*x +5*x^2 +16*x^3 +14*x^4 +24*x^5 +18*x^6 +20*x^7 +5*x^8 + x^10 -4*x^11 +4*x^12)/((1 -x)^4*(1 +x)*(1 +x^2)^2*(1 +x +x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 3*a(n-3) - 2*a(n-4) - 2*a(n-7) + 3*a(n-8) - 2*a(n-9) + 2*a(n-10) - a(n-11) for n>12.
(End)

A299268 Coordination sequence for "crs" 3D uniform tiling formed from tetrahedra and truncated tetrahedra.

Original entry on oeis.org

1, 6, 18, 48, 78, 126, 182, 240, 330, 390, 522, 576, 758, 798, 1038, 1056, 1362, 1350, 1730, 1680, 2142, 2046, 2598, 2448, 3098, 2886, 3642, 3360, 4230, 3870, 4862, 4416, 5538, 4998, 6258, 5616, 7022, 6270, 7830, 6960, 8682, 7686, 9578, 8448, 10518, 9246

Offset: 0

N. J. A. Sloane, Feb 07 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 #6.

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

See A299269 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.

I:=[18, 48, 78, 126, 182, 240, 330]; [1,6] cat [n le 6 select I[n] else 3*Self(n-2) -3*Self(n-4) + Self(n-6): n in [1..30]]; // G. C. Greubel, Feb 20 2018

CoefficientList[Series[(x^6+27*x^4+30*x^3+15*x^2+6*x+1)/(1-x^2)^3, {x, 0, 50}], x] (* G. C. Greubel, Feb 20 2018 *)

Vec((1 + 6*x + 15*x^2 + 30*x^3 + 27*x^4 + x^6) / ((1 - x)^3*(1 + x)^3) + O(x^60)) \\ Colin Barker, Feb 09 2018

G.f.: (x^6 + 27*x^4 + 30*x^3 + 15*x^2 + 6*x + 1) / (1 - x^2)^3.
From Colin Barker, Feb 09 2018: (Start)
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
a(n) = (11*n^2 - 6*n + 4) / 2 for n>0 and even.
a(n) = 3 * (3*n^2 + 2*n - 1) / 2 for n odd. (End)
E.g.f.: ((11*x^2 + 15*x + 4)*cosh(x) + (9*x^2 + 5*x - 3)*sinh(x) - 2)/2. - Stefano Spezia, Mar 14 2024

A299269 Partial sums of A299268.

Original entry on oeis.org

1, 7, 25, 73, 151, 277, 459, 699, 1029, 1419, 1941, 2517, 3275, 4073, 5111, 6167, 7529, 8879, 10609, 12289, 14431, 16477, 19075, 21523, 24621, 27507, 31149, 34509, 38739, 42609, 47471, 51887, 57425, 62423, 68681, 74297, 81319, 87589, 95419, 102379, 111061

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 (1,3,-3,-3,3,1,-1).

Cf. A299268.

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:=[25,73,151,277,459,699,1029]; [1,7] cat [n le 7 select I[n] else Self(n-1) + 3*Self(n-2) - 3*Self(n-3) - 3*Self(n-4) + 3*Self(n-5) + Self(n-6) - Self(n-7): n in [1..30]]; // G. C. Greubel, Feb 20 2018

CoefficientList[Series[(1+6*x+15*x^2+30*x^3+27*x^4+x^6)/((1-x)^4*(1+ x)^3), {x, 0, 50}], x] (* G. C. Greubel, Feb 20 2018 *)

Vec((1 + 6*x + 15*x^2 + 30*x^3 + 27*x^4 + x^6) / ((1 - x)^4*(1 + x)^3) + O(x^60)) \\ Colin Barker, Feb 09 2018

From Colin Barker, Feb 09 2018: (Start)
G.f.: (1 + 6*x + 15*x^2 + 30*x^3 + 27*x^4 + x^6) / ((1 - x)^4*(1 + x)^3).
a(n) = (20*n^3 + 33*n^2 - 2*n + 12) / 12 for n even.
a(n) = (20*n^3 + 27*n^2 + 28*n + 9) / 12 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>6. (End)
E.g.f.: ((12 + 75*x + 93*x^2 + 20*x^3)*cosh(x) + (9 + 51*x + 87*x^2 + 20*x^3)*sinh(x))/12. - Stefano Spezia, Mar 14 2024

A299272 Coordination sequence for "flu" 3D uniform tiling formed from tetrahedra, rhombicuboctahedra, and cubes.

Original entry on oeis.org

1, 6, 18, 37, 63, 99, 142, 189, 249, 317, 384, 468, 562, 648, 756, 877, 981, 1113, 1262, 1383, 1539, 1717, 1854, 2034, 2242, 2394, 2598, 2837, 3003, 3231, 3502, 3681, 3933, 4237, 4428, 4704, 5042, 5244, 5544, 5917, 6129, 6453, 6862, 7083, 7431, 7877, 8106, 8478, 8962, 9198

Offset: 0

N. J. A. Sloane, Feb 10 2018

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

The tiling is called "3-RCO-trille" in Conway, Burgiel, Goodman-Strauss, 2008, p. 297. - Felix Fröhlich, Feb 11 2018

J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5.
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #5.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Reticular Chemistry Structure Resource (RCSR), The flu tiling (or net)
Wikipedia, Tetrahedral-octahedral honeycomb - Runcic cubic honeycomb
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

See A299273 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.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3/(1-x^3)^3)); // G. C. Greubel, Feb 20 2018

CoefficientList[Series[(x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3/(1-x^3)^3, {x, 0, 50}], x] (* G. C. Greubel, Feb 20 2018 *)

x='x+O('x^30); Vec((x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3/(1-x^3)^3) \\ G. C. Greubel, Feb 20 2018

Conjectures from Colin Barker, Feb 11 2018: (Start)
G.f.: (1 + x)^3*(1 + x^2)*(1 + 3*x + 5*x^2 + 3*x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)^3).
a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n>9.
(End)
G.f.: (x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3 / (1-x^3)^3. - N. J. A. Sloane, Feb 12 2018 (This confirms my conjecture from Feb 10 2018 and the above conjecture from Colin Barker.)
a(n) = (60 + 104*n^2 + (n^2 - 6)*cos(2*n*Pi/3) - 3*sqrt(3)*n*sin(2*n*Pi/3))/27 for n > 0. - Stefano Spezia, Jan 23 2022

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

cp's OEIS Frontend

A299260 Partial sums of A299254.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A299261 Partial sums of A299255.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A299263 Partial sums of A299257.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A299264 Partial sums of A299258.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A299265 Partial sums of A299259.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A299266 Coordination sequence for "cab" 3D uniform tiling formed from octahedra and truncated cubes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A299267 Partial sums of A299266.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma