A299254 -id:A299254 - cp's OEIS frontend

A005901 Number of points on surface of cuboctahedron (or icosahedron): a(0) = 1; for n > 0, a(n) = 10n^2 + 2. Also coordination sequence for f.c.c. or A_3 or D_3 lattice.

1, 12, 42, 92, 162, 252, 362, 492, 642, 812, 1002, 1212, 1442, 1692, 1962, 2252, 2562, 2892, 3242, 3612, 4002, 4412, 4842, 5292, 5762, 6252, 6762, 7292, 7842, 8412, 9002, 9612, 10242, 10892, 11562, 12252, 12962, 13692, 14442, 15212, 16002

Offset: 0

N. J. A. Sloane, R. Vaughan

Sequence found by reading the segment (1, 12) together with the line from 12, in the direction 12, 42, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012

H. S. M. Coxeter, "Polyhedral numbers," in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
Gmelin Handbook of Inorg. and Organomet. Chem., 8th Ed., 1994, TYPIX search code (225) cF4
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #1.
R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.
S. Rosen, Wizard of the Dome: R. Buckminster Fuller; Designer for the Future. Little, Brown, Boston, 1969, p. 109.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
R. W. Grosse-Kunstleve, Coordination Sequences and Encyclopedia of Integer Sequences
R. W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, Algebraic Description of Coordination Sequences and Exact Topological Densities for Zeolites, Acta Cryst., A52 (1996), pp. 879-889.
G. Nebe and N. J. A. Sloane, Home page for this lattice
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Reticular Chemistry Structure Resource (RCSR), The fcu tiling (or net)
N. J. A. Sloane, A portion of the f.c.c. lattice packing.
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
K. Urner, Microarchitecture of the Virus
R. Vaughan & N. J. A. Sloane, Correspondence, 1975
Wikipedia, Cuboctahedron
Index entries for sequences related to f.c.c. lattice
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000290, A004015, A027599.
Cf. A033583, A062786, A092277, A206399.
Partial sums give A005902.
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.

[n eq 0 select 1 else 2*(5*n^2+1): n in [0..55]]; // G. C. Greubel, May 25 2023

Join[{1},10*Range[40]^2+2] (* or *) Join[{1},LinearRecurrence[{3,-3,1},{12,42,92},40]] (* Harvey P. Dale, May 28 2014 *)

PARI
```
a(n)=if(n<0,0,10*n^2+1+(n>0))
```

[2*(5*n^2 + 1)-int(n==0) for n in range(56)] # G. C. Greubel, May 25 2023

G.f.: (1+x)*(1+8*x+x^2)/(1-x)^3. - Simon Plouffe in his 1992 dissertation
G.f. for coordination sequence for A_n lattice is (1-z)^(-n) *  Sum_{i=0..n} binomial(n, i)^2*z^i. [Bacher et al.]
a(n+1) = A027599(n+2) + A092277(n+1) - Creighton Dement, Feb 11 2005
a(n) = 2 + A033583(n), n >= 1. - Omar E. Pol, Jul 18 2012
a(n) = 12 + 24*(n-1) + 8*A000217(n-2) + 6*A000290(n-1). The properties of the cuboctahedron, namely, its number of vertices (12), edges (24), and faces as well as face-type (8 triangles and 6 squares), are involved in this formula. - Peter M. Chema, Mar 26 2017
a(n) = A062786(n) + A062786(n+1). - R. J. Mathar, Feb 28 2018
E.g.f.: -1 + 2*(1 + 5*x + 5*x^2)*exp(x). - G. C. Greubel, May 25 2023
Sum{n>=0} 1/a(n) = 3/4 + Pi*sqrt(5)*coth(Pi/sqrt 5)/20 = 1.14624... - R. J. Mathar, Apr 27 2024

A007202 Crystal ball sequence for hexagonal close-packing.

Original entry on oeis.org

1, 13, 57, 153, 323, 587, 967, 1483, 2157, 3009, 4061, 5333, 6847, 8623, 10683, 13047, 15737, 18773, 22177, 25969, 30171, 34803, 39887, 45443, 51493, 58057, 65157, 72813, 81047, 89879, 99331, 109423, 120177, 131613, 143753, 156617

Offset: 0

N. J. A. Sloane and J. H. Conway

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Xiaogang Liang, Ilyar Hamid, and Haiming Duan, Dynamic stabilities of icosahedral-like clusters and their ability to form quasicrystals, AIP Advances 6, 065017 (2016).
Index entries for crystal ball sequences
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Partial sums of A007899.

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.

Table[Floor[(7((n+1)^4-n^4)+4)/8],{n,0,40}] (* or *) LinearRecurrence[ {3,-2,-2,3,-1},{1,13,57,153,323},40] (* Harvey P. Dale, Jul 15 2011 *)

j=[]; for(n=0,75,j=concat(j,round((7/8)*((n+1)^4-n^4)))); j

def a(n): return round((7/8)*((n+1)**4-n**4))
print([a(n) for n in range(36)]) # Michael S. Branicky, Jan 13 2021

Nearest integer to (7/8)*( (n+1)^4 - n^4 ).
G.f.: (x^4+10*x^3+20*x^2+10*x+1)/(x-1)^4/(x+1).
a(n) = 7*(2*n+1)*(2*n^2+2*n+1)/8 +(-1)^n/8. - R. J. Mathar, Mar 24 2011
a(0)=1, a(1)=13, a(2)=57, a(3)=153, a(4)=323, a(n)=3*a(n-1)- 2*a(n-2)- 2*a(n-3)+3*a(n-4)-a(n-5). - Harvey P. Dale, Jul 15 2011
E.g.f.: ((4 + 49*x + 63*x^2 + 14*x^3)*cosh(x) + (3 + 49*x + 63*x^2+ 14*x^3)*sinh(x))/4. - Stefano Spezia, Mar 14 2024

More terms from Jason Earls, Jul 14 2001

A007899 Coordination sequence for hexagonal close-packing.

Original entry on oeis.org

1, 12, 44, 96, 170, 264, 380, 516, 674, 852, 1052, 1272, 1514, 1776, 2060, 2364, 2690, 3036, 3404, 3792, 4202, 4632, 5084, 5556, 6050, 6564, 7100, 7656, 8234, 8832, 9452, 10092, 10754, 11436, 12140, 12864, 13610, 14376, 15164, 15972, 16802, 17652, 18524, 19416, 20330, 21264, 22220

Offset: 0

N. J. A. Sloane and J. H. Conway

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #2.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Reticular Chemistry Structure Resource (RCSR), The hcp tiling (or net)
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

For partial sums see A007202.

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:=[1,12,44,96,170]; [n le 5 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Feb 16 2014

[1] cat [2 + Floor(21*n^2/2): n in [1..50]]; // G. C. Greubel, Feb 20 2018

Join[{1},Floor[(21Range[40]^2)/2]+2] (* or *) Join[{1},LinearRecurrence[ {2,0,-2,1},{12,44,96,170},40]] (* Harvey P. Dale, Feb 15 2014 *)
CoefficientList[Series[(x^4 + 10 x^3 + 20 x^2 + 10 x + 1)/(1 - x)^3/(x + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 16 2014 *)

for(n=0,50, print1(if(n==0,1, 2 + floor(21*n^2/2)), ", ")) \\ G. C. Greubel, Feb 20 2018

a(n) = floor( 21*n^2 / 2 ) + 2, for n>= 1.
G.f.: (x^4 +10*x^3 +20*x^2 +10*x +1)/((1+x)*(1-x)^3).
a(0)=1, a(1)=12, a(2)=44, a(3)=96, a(4)=170, a(n)=2*a(n-1)-2*a(n-3)+ a(n-4). - Harvey P. Dale, Feb 15 2014
a(n) = (21/2)*n^2 + 7/4 + (1/4)*(-1)^n - 0^n. - Eric Simon Jacob, Feb 12 2023
E.g.f.: ((4 + 21*x + 21*x^2)*cosh(x) + 3*(1 + 7*x + 7*x^2)*sinh(x) - 2)/2. - Stefano Spezia, Mar 14 2024

A010001 a(0) = 1, a(n) = 5*n^2 + 2 for n>0.

Original entry on oeis.org

1, 7, 22, 47, 82, 127, 182, 247, 322, 407, 502, 607, 722, 847, 982, 1127, 1282, 1447, 1622, 1807, 2002, 2207, 2422, 2647, 2882, 3127, 3382, 3647, 3922, 4207, 4502, 4807, 5122, 5447, 5782, 6127, 6482, 6847, 7222, 7607, 8002, 8407, 8822, 9247, 9682, 10127, 10582

Offset: 0

N. J. A. Sloane

Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3^3.4^2 2D tiling (cf. A008706). - N. J. A. Sloane, Feb 07 2018

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #13.

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

Cf. A008706, A206399.
See A063489 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.

lst={};Do[AppendTo[lst,5*n^2+2],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Join[{1}, 5 Range[46]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)

A010001(n)=5*n^2+2-!n   \\ M. F. Hasler, Feb 14 2012

G.f.: (1+x)*(1+3*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*5+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+sqrt(10)/20*Pi*coth( Pi/5 *sqrt 10) = 1.2657655... - R. J. Mathar, May 07 2024

More terms from Bruno Berselli, Feb 06 2012

A299279 Coordination sequence for "reo" 3D uniform tiling.

Original entry on oeis.org

1, 8, 30, 68, 126, 180, 286, 348, 510, 572, 798, 852, 1150, 1188, 1566, 1580, 2046, 2028, 2590, 2532, 3198, 3092, 3870, 3708, 4606, 4380, 5406, 5108, 6270, 5892, 7198, 6732, 8190, 7628, 9246, 8580, 10366, 9588, 11550, 10652, 12798, 11772, 14110, 12948, 15486, 14180

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 #7.

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

See A299280 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[{0, 3, 0, -3, 0, 1}, {1, 8, 30, 68, 126, 180, 286, 348}, 50] (* Paolo Xausa, Jan 16 2025 *)

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

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

A299255 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.3.4.3.4 2D tiling (cf. A219529).

Original entry on oeis.org

1, 7, 23, 50, 87, 135, 194, 263, 343, 434, 535, 647, 770, 903, 1047, 1202, 1367, 1543, 1730, 1927, 2135, 2354, 2583, 2823, 3074, 3335, 3607, 3890, 4183, 4487, 4802, 5127, 5463, 5810, 6167, 6535, 6914, 7303, 7703, 8114, 8535, 8967, 9410, 9863, 10327, 10802, 11287, 11783, 12290, 12807, 13335

Offset: 0

N. J. A. Sloane, Feb 07 2018

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #14.

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

Cf. A219529.
See A299261 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. A099837.

LinearRecurrence[{2,-1,1,-2,1},{1,7,23,50,87,135},60] (* Harvey P. Dale, Apr 01 2018 *)

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

G.f.: (x + 1)^5 / ((x^2 + x + 1)*(1 - x)^3).
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) = 2*(8 + 24*n^2 + A099837(n+3)/2)/9 for n > 0. - Stefano Spezia, Jun 06 2024

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

Original entry on oeis.org

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

cp's OEIS Frontend

A005901 Number of points on surface of cuboctahedron (or icosahedron): a(0) = 1; for n > 0, a(n) = 10n^2 + 2. Also coordination sequence for f.c.c. or A_3 or D_3 lattice.

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A007202 Crystal ball sequence for hexagonal close-packing.

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A007899 Coordination sequence for hexagonal close-packing.

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A010001 a(0) = 1, a(n) = 5*n^2 + 2 for n>0.

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A299279 Coordination sequence for "reo" 3D uniform tiling.

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A299255 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.3.4.3.4 2D tiling (cf. A219529).

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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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