A005897 -id:A005897 - cp's OEIS frontend

A299289 Coordination sequence for "tsi" 3D uniform tiling.

1, 8, 28, 60, 106, 164, 236, 320, 418, 528, 652, 788, 938, 1100, 1276, 1464, 1666, 1880, 2108, 2348, 2602, 2868, 3148, 3440, 3746, 4064, 4396, 4740, 5098, 5468, 5852, 6248, 6658, 7080, 7516, 7964, 8426, 8900, 9388, 9888, 10402, 10928

Offset: 0

N. J. A. Sloane, Feb 10 2018

nonn

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

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #12.

Reticular Chemistry Structure Resource (RCSR), The tsi tiling (or net)

See A299290 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.

Conjectures from Colin Barker, Feb 11 2018: (Start)
G.f.: (1 + 6*x + 12*x^2 + 6*x^3 + x^4) / ((1 - x)^3*(1 + x)).
a(n) = (13*n^2 + 4) / 2 for n>0 and even.
a(n) = (13*n^2 + 3) / 2 for n odd.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>4. (End)
Conjectured e.g.f.: ((4 + 13*x + 13*x^2)*cosh(x) + (3 + 13*x + 13*x^2)*sinh(x) - 2)/2. - Stefano Spezia, Jun 08 2024

A299290 Partial sums of A299289.

Original entry on oeis.org

1, 9, 37, 97, 203, 367, 603, 923, 1341, 1869, 2521, 3309, 4247, 5347, 6623, 8087, 9753, 11633, 13741, 16089, 18691, 21559, 24707, 28147, 31893, 35957, 40353, 45093, 50191, 55659, 61511, 67759, 74417, 81497, 89013, 96977, 105403, 114303, 123691

Offset: 0

N. J. A. Sloane, Feb 10 2018

nonn

Cf. A299289.

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.

Conjectures from Colin Barker, Feb 11 2018: (Start)
G.f.: (1 + 6*x + 12*x^2 + 6*x^3 + x^4) / ((1 - x)^4*(1 + x)).
a(n) = (12 + 34*n + 39*n^2 + 26*n^3) / 12 for n even.
a(n) = (9 + 34*n + 39*n^2 + 26*n^3) / 12 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>4.
(End)

A299291 Coordination sequence for "ubt" 3D uniform tiling.

Original entry on oeis.org

1, 5, 14, 29, 56, 85, 130, 181, 226, 299, 382, 445, 538, 635, 708, 845, 962, 1079, 1218, 1363, 1456, 1671, 1808, 1987, 2170, 2365, 2470, 2777, 2920, 3169, 3394, 3641, 3750, 4163, 4298, 4625, 4890, 5191, 5296, 5829, 5942, 6355, 6658, 7015, 7108, 7775, 7852, 8359, 8698, 9113, 9186

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 #10.

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

See A299292 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[{-1,0,1,1,0,2,2,0,-2,-2,0,-1,-1,0,1,1},{1,5,14,29,56,85,130,181,226,299,382,445,538,635,708,845,962,1079,1218,1363,1456},60] (* Harvey P. Dale, Aug 20 2021 *)

Vec((12*x^20 + 16*x^19 - 20*x^17 - 27*x^16 - 8*x^15 + 3*x^14 + 46*x^13 + 115*x^12 + 176*x^11 + 212*x^10 + 226*x^9 + 228*x^8 + 214*x^7 + 170*x^6 + 122*x^5 + 79*x^4 + 42*x^3 + 19*x^2 + 6*x + 1) / ((1 + x)*(1 - x^3)*(1 - x^6)^2) + O(x^50)) \\ Colin Barker, Feb 14 2018

G.f.: (12*x^20 + 16*x^19 - 20*x^17 - 27*x^16 - 8*x^15 + 3*x^14 + 46*x^13 + 115*x^12 + 176*x^11 + 212*x^10 + 226*x^9 + 228*x^8 + 214*x^7 + 170*x^6 + 122*x^5 + 79*x^4 + 42*x^3 + 19*x^2 + 6*x + 1) / ((1 + x)*(1 - x^3)*(1 - x^6)^2). - N. J. A. Sloane, Feb 13 2018

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

A299292 Partial sums of A299291.

Original entry on oeis.org

1, 6, 20, 49, 105, 190, 320, 501, 727, 1026, 1408, 1853, 2391, 3026, 3734, 4579, 5541, 6620, 7838, 9201, 10657, 12328, 14136, 16123, 18293, 20658, 23128, 25905, 28825, 31994, 35388, 39029, 42779, 46942, 51240, 55865, 60755, 65946, 71242, 77071, 83013, 89368, 96026

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
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).

Cf. A299291.

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((12*x^20 + 16*x^19 - 20*x^17 - 27*x^16 - 8*x^15 + 3*x^14 + 46*x^13 + 115*x^12 + 176*x^11 + 212*x^10 + 226*x^9 + 228*x^8 + 214*x^7 + 170*x^6 + 122*x^5 + 79*x^4 + 42*x^3 + 19*x^2 + 6*x + 1) / ((1 - x^2)*(1 - x^3)*(1 - x^6)^2) + O(x^50)) \\ Colin Barker, Feb 14 2018

G.f.: (12*x^20 + 16*x^19 - 20*x^17 - 27*x^16 - 8*x^15 + 3*x^14 + 46*x^13 + 115*x^12 + 176*x^11 + 212*x^10 + 226*x^9 + 228*x^8 + 214*x^7 + 170*x^6 + 122*x^5 + 79*x^4 + 42*x^3 + 19*x^2 + 6*x + 1) / ((1 - x^2)*(1 - x^3)*(1 - x^6)^2).

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>17. - Colin Barker, Feb 14 2018

A206399 a(0) = 1; for n > 0, a(n) = 41*n^2 + 2.

Original entry on oeis.org

1, 43, 166, 371, 658, 1027, 1478, 2011, 2626, 3323, 4102, 4963, 5906, 6931, 8038, 9227, 10498, 11851, 13286, 14803, 16402, 18083, 19846, 21691, 23618, 25627, 27718, 29891, 32146, 34483, 36902, 39403, 41986, 44651, 47398, 50227, 53138, 56131, 59206, 62363, 65602

Offset: 0

Bruno Berselli, Feb 07 2012

Apart from the first term, numbers of the form (r^2 + 2*s^2)*n^2 + 2 = (r*n)^2 + (s*n - 1)^2 + (s*n + 1)^2: in this case is r = 3, s = 4. After 1, all terms are in A000408.

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

Sequences of the same type: A005893, A005897, A005899, A005901, A005903, A005905, A005914, A005918, A005919, A008527, A010000-A010023.

Cf. A000408.

[n eq 0 select 1 else 41*n^2+2: n in [0..39]];

I:=[1,43,166,371]; [n le 4 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..41]]; // Vincenzo Librandi, Aug 18 2013

Join[{1}, 41 Range[39]^2 + 2]
CoefficientList[Series[(1 + x) (1 + 39 x + x^2) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

a(n)=if(n,41*n^2+2,1) \\ Charles R Greathouse IV, Sep 24 2015

O.g.f.: (1 + x)*(1 + 39*x + x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. - Wesley Ivan Hurt, Dec 18 2020
E.g.f.: exp(x)*(41*x^2 + 41*x + 2) - 1. - Elmo R. Oliveira, Nov 29 2024

A005914 Number of points on surface of hexagonal prism: 12*n^2 + 2 for n > 0 (coordination sequence for W(2)).

Original entry on oeis.org

1, 14, 50, 110, 194, 302, 434, 590, 770, 974, 1202, 1454, 1730, 2030, 2354, 2702, 3074, 3470, 3890, 4334, 4802, 5294, 5810, 6350, 6914, 7502, 8114, 8750, 9410, 10094, 10802, 11534, 12290, 13070, 13874, 14702, 15554, 16430, 17330, 18254, 19202, 20174, 21170

Offset: 0

N. J. A. Sloane and Ralf W. Grosse-Kunstleve

For n >= 1, a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n,n+1} such that Im(f) contains 2 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Feb 24 2007
Equals binomial transform of [1, 13, 23, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 22 2008
First bisection of A005918. After 1, all terms are in A000408 (see Formula section). - Bruno Berselli, Feb 07 2012
Also sequence found by reading the segment (1, 14) together with the line from 14, in the direction 14, 50, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Nov 02 2012
Unique sequence such that for all n > 0, n*a(1) + (n-1)*a(2) + (n-3)*a(3) + ... + 2*a(2) + a(1) = n^4. - Warren Breslow, Dec 12 2014

Gmelin Handbook of Inorganic and Organometallic Chemistry, 8th Ed., 1994, TYPIX search code (229) cI2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Ovidiu Bagdasar, On Some Functions Involving the lcm and gcd of Integer Tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, No. 2 (2014), pp. 91-100.
Ralf W. Grosse-Kunstleve, Coordination Sequences and Encyclopedia of Integer Sequences, 1996.
Ralf 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.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Claudio de J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq., Vol. 16 (2013), Article 13.5.7.
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.
Boon K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem., Vol. 24 (1985), pp. 4545-4558.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

First differences of A005917.

Cf. A000408, A001082, A003154, A005918, A206399.

A005914:=-(z+1)*(z**2+10*z+1)/(z-1)**3; # Simon Plouffe in his 1992 dissertation.

Table[If[n == 0, 1, 12*n^2 + 2], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
Join[{1},LinearRecurrence[{3,-3,1},{14,50,110},50]] (* Harvey P. Dale, Oct 09 2012 *)

a(n)=12*n^2+2 \\ Charles R Greathouse IV, Jan 31 2012

G.f.: (1+x)*(1+10*x+x^2)/(1-x)^3. - Simon Plouffe (see MAPLE line)
a(n) = (2n-1)^2 + (2n)^2 + (2n+1)^2 for n > 0. - Bruno Berselli, Jan 30 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=14, a(2)=50, a(3)=110. - Harvey P. Dale, Oct 09 2012
E.g.f.: exp(x)*(12*x^2 + 12*x + 2) - 1. - Alois P. Heinz, Sep 10 2013
From Bruce J. Nicholson, Jan 19 2019: (Start)
Sum_{i=1..n} a(i) = A005917(n+1).
a(n) = A003154(n) + A003154(n+1). (End)
From Amiram Eldar, Jan 27 2022: (Start)
Sum_{n>=0} 1/a(n) = ((Pi/sqrt(6))*coth(Pi/sqrt(6)) + 3)/4.
Sum_{n>=0} (-1)^n/a(n) = ((Pi/sqrt(6))*cosech(Pi/sqrt(6)) + 3)/4. (End)

A010014 a(0) = 1, a(n) = 24*n^2 + 2 for n>0.

Original entry on oeis.org

1, 26, 98, 218, 386, 602, 866, 1178, 1538, 1946, 2402, 2906, 3458, 4058, 4706, 5402, 6146, 6938, 7778, 8666, 9602, 10586, 11618, 12698, 13826, 15002, 16226, 17498, 18818, 20186, 21602, 23066, 24578, 26138, 27746, 29402, 31106, 32858, 34658, 36506, 38402, 40346

Offset: 0

N. J. A. Sloane

Number of points of L_infinity norm n in the simple cubic lattice Z^3. - N. J. A. Sloane, Apr 15 2008
Numbers of cubes needed to completely "cover" another cube. - Xavier Acloque, Oct 20 2003
First bisection of A005897. After 1, all terms are in A000408. - Bruno Berselli, Feb 06 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Xavier Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Victor Kamel, Hanxueyu Yan, and Sean Chester, CLOVER: A GPU-native, Spatio-graph-based Approach to Exact kNN, ACM Int'l Conf. Supercomp. (ICS 2025). See p. 3.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399.

Join[{1}, 24 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)

a(n) = if (n==0, 1, 24*n^2 + 2);
vector(40, n, a(n-1)) \\ Altug Alkan, Sep 29 2015

a(n) = (2*n+1)^3 - (2*n-1)^3 for n >= 1. - Xavier Acloque, Oct 20 2003
G.f.: (1+x)*(1+22*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
a(n) = (2*n-1)^2 + (2*n+1)^2 + (4*n)^2 for n>0. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*24+2)*exp(x)-1. - Gopinath A. R., Feb 14 2012
a(n) = A005899(n) + A195322(n), n > 0. - R. J. Cano, Sep 29 2015
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(3)/24*Pi*coth(Pi*sqrt(3)/6) = 1.065052868574... - R. J. Mathar, May 07 2024
a(n) = 2*A158480(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A069190(n)+A069190(n+1). - R. J. Mathar, May 07 2024

More terms from Xavier Acloque, Oct 20 2003

A009927 Coordination sequence for Cr3Si, Si position.

Original entry on oeis.org

1, 12, 50, 120, 218, 344, 546, 728, 902, 1212, 1526, 1784, 2154, 2552, 2954, 3432, 3854, 4340, 4998, 5504, 6002, 6768, 7442, 8024, 8814, 9572, 10334, 11232, 11978, 12824, 13938, 14768, 15590, 16812, 17846, 18752, 19962, 21080, 22202, 23520

Offset: 0

Ralf W. Grosse-Kunstleve

nonn

Gmelin Handbook of Inorg. and Organomet. Chem., 8th Ed., 1994, TYPIX search code (223) cP8.

R. W. Grosse-Kunstleve, 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. See Table I.
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.f.: (1+12*x+51*x^2+130*x^3+243*x^4+350*x^5+450*x^6+418*x^7 +327*x^8+182*x^9+51*x^10+16*x^11-7*x^12+8*x^13+12*x^14)/ ((1+x)*(1+x^2)^2*(1+x+x^2)^2*(1-x)^3). - Robert Israel, Dec 18 2015
Empirical: a(n) = (1903/72) + (3/8)*(-1)^n + 19*KroneckerDelta[n,0] - 8*KroneckerDelta[n,1] - 12*KroneckerDelta[n,2] + ((n+1)/12)*(187*n-273) - (32*sqrt(3)/27)*((13/2)*cos((4n+1)*Pi/6) + sin(2n*Pi/3)) - (3*sqrt(26)/2)*(-1)^n*cos(n*Pi/2 + arctan(1/5)) - (3/4)*i^n*(1+(-1)^n)*(n+2). - G. C. Greubel, Dec 18 2015
G.f.: (1 + 12*x + 50*x^2 + 118*x^3 + 192*x^4 + 220*x^5 + 207*x^6 + 68*x^7-123*x^8-236*x^9-276*x^10-166*x^11-58*x^12-8*x^13 + 19*x^14-8*x^15-12*x^16) / (1-x^3)^2 / (1-x^4)^2. - Sean A. Irvine, Mar 15 2018

A008084 Coordination sequence T1 for Zeolite Code ACO, ASV, EDI, and THO.

Original entry on oeis.org

1, 4, 9, 19, 35, 52, 72, 100, 131, 163, 201, 244, 290, 340, 393, 451, 515, 580, 648, 724, 803, 883, 969, 1060, 1154, 1252, 1353, 1459, 1571, 1684, 1800, 1924, 2051, 2179, 2313, 2452, 2594, 2740, 2889, 3043, 3203, 3364, 3528, 3700, 3875, 4051, 4233, 4420

Offset: 0

Ralf W. Grosse-Kunstleve

W. M. Meier, D. H. Olson, and Ch. Baerlocher, Atlas of Zeolite Structure Types, 4th Ed., Elsevier, 1996.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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.
Sean A. Irvine, Generating Functions for Coordination Sequences of Zeolites, after Grosse-Kunstleve, Brunner, and Sloane.
International Zeolite Association, Database of Zeolite Structures.
Reticular Chemistry Structure Resource, nets pcb, pcb-b, asv, edi, edi-c.
Index entries for linear recurrences with constant coefficients, signature (2,-2,3,-3,2,-2,1).

CoefficientList[Series[-(x + 1)^3 (x^4 - x^3 + 3 x^2 - x + 1)/((x - 1)^3 (x^2 + 1) (x^2 + x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2013 *)

For n > 1, a(n) = 2n^2 - 4n + 4 + p(n), with the 12-periodic sequence p(n) with period {0, 0, 0, -1, -1, 1, 0, -2, 0, 1, -1, -1}.
a(12*m+k) = 288*m^2 + 48*k*m + [ 2, 4, 9, 19, 35, 52, 72, 100, 131, 163, 201, 244 ], 0 <= k < 12. - N. J. A. Sloane
G.f.: -(x+1)^3*(x^4-x^3+3*x^2-x+1) / ((x-1)^3*(x^2+1)*(x^2+x+1)). - Colin Barker, Dec 12 2012

A008264 Coordination sequence for tridymite, lonsdaleite, and wurtzite.

Original entry on oeis.org

1, 4, 12, 25, 44, 67, 96, 130, 170, 214, 264, 319, 380, 445, 516, 592, 674, 760, 852, 949, 1052, 1159, 1272, 1390, 1514, 1642, 1776, 1915, 2060, 2209, 2364, 2524, 2690, 2860, 3036, 3217, 3404, 3595, 3792, 3994, 4202, 4414, 4632, 4855, 5084, 5317, 5556, 5800

Offset: 0

Ralf W. Grosse-Kunstleve

Inorganic Crystal Structure Database: Collection Code 29343
Michael O'Keeffe, Topological and geometrical characterization of sites in silicon carbide polytypes, Chemistry of Materials 3 (2) (1991), 332-335. (Eq. (2) gives an empirical formula for a(n). - N. J. A. Sloane, Apr 07 2018)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. L. Glasser, Symmetry properties of the wurtzite structure, Journal of Physics and Chemistry of Solids, 10(2-3) (1959), 229-241.
Ralf W. Grosse-Kunstleve, Zeolites, Frameworks, Coordination Sequences and Encyclopedia of Integer Sequences, 1996.
Ralf 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), 879-889.
Sean A. Irvine, Generating Functions for Coordination Sequences of Zeolites, after Grosse-Kunstleve, Brunner, and Sloane.
Michael O'Keeffe, N-dimensional diamond, sodalite and rare sphere packings, Acta Cryst. A 47 (1991), 749-753.
Michael O'Keeffe, Topological and geometrical characterization of sites in silicon carbide polytypes, Chemistry of Materials 3 (2) (1991), 332-335.
Reticular Chemistry Structure Resource (RCSR), The lon net (lonsdaleite) and The lon-b net (wurtzite).
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A008524 for 4-D analog, A008253 for diamond.

Cf. A217511 for theta series.

a[n_] := (m = Quotient[n, 4]; k = Mod[n, 4]; 42*m^2 + 21*k*m + Switch[k, 0, 2, 1, 4, 2, 12, 3, 25]); a[0]=1; Table[a[n], {n, 0, 47}] (* Jean-François Alcover, Oct 11 2012, from the first formula *)
Join[{1}, Table[1 + (42 n^2 + (1 + (-1)^n) (3 + 2 (-1)^((n - 1) n/2)) + 6)/16, {n, 50}]] (* Bruno Berselli, Jul 24 2013 *)
LinearRecurrence[{2,-1,0,1,-2,1},{1,4,12,25,44,67,96},20] (* Harvey P. Dale, Dec 27 2016 *)

a(n)=if(n, 1+(42*n^2+(1+(-1)^n)*(3+2*(-1)^((n-1)*n/2))+6)/16, 1) \\ Charles R Greathouse IV, Feb 10 2017

a(4*m+k) = 42*m^2 + 21*k*m + [ 2, 4, 12, 25 ], 0 <= k < 4 (N. J. A. Sloane).
a(n) = 1 + (42*n^2 + (1 + (-1)^n)*(3 + 2*(-1)^((n - 1)*n/2)) + 6)/16 for n > 0, a(0) = 1. - Bruno Berselli, Jul 24 2013
G.f.: (1 + 2*x + 5*x^2 + 5*x^3 + 5*x^4 + 2*x^5 + x^6)/((1 - x)^3*(1 + x + x^2 + x^3)). - Bruno Berselli, Jul 24 2013

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