A001845 -id:A001845 - cp's OEIS frontend

A005893 Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).

1, 4, 10, 20, 34, 52, 74, 100, 130, 164, 202, 244, 290, 340, 394, 452, 514, 580, 650, 724, 802, 884, 970, 1060, 1154, 1252, 1354, 1460, 1570, 1684, 1802, 1924, 2050, 2180, 2314, 2452, 2594, 2740, 2890, 3044, 3202, 3364, 3530, 3700, 3874, 4052, 4234

Offset: 0

N. J. A. Sloane

Number of n-matchings of the wheel graph W_{2n} (n > 0). Example: a(2)=10 because in the wheel W_4 (rectangle ABCD and spokes OA,OB,OC,OD) we have the 2-matchings: (AB, OC), (AB, OD), (BC, OA), (BC,OD), (CD,OA), (CD,OB), (DA,OB), (DA,OC), (AB,CD) and (BC,DA). - Emeric Deutsch, Dec 25 2004
For n > 0 a(n) is the difference of two tetrahedral (or pyramidal) numbers: binomial(n+3, 3) = (n+1)(n+2)(n+3)/6. a(n) = A000292(n+1) - A000292(n-3) = (n+1)(n+2)(n+3)/6 - (n-3)(n-2)(n-1)/6. - Alexander Adamchuk, May 20 2006; updated by Peter Munn, Aug 25 2017 due to changed offset in A000292
Equals binomial transform of [1, 3, 3, 1, -1, 1, -1, 1, -1, 1, ...]. Binomial transform of A005893 = nonzero terms of A053545: (1, 5, 19, 63, 191, ...). - Gary W. Adamson, Apr 28 2008
Disregarding the terms < 10, the sums of four consecutive triangular numbers (A000217). - Rick L. Shepherd, Sep 30 2009
Use a set of n concentric circles where n >= 0 to divide the plane. a(n) is the maximal number of regions after the 2nd division. - Frank M Jackson, Sep 07 2011
Euler transform of length 4 sequence [4, 0, 0, -1]. - Michael Somos, May 14 2014
Also, growth series for affine Coxeter group (or affine Weyl group) A_3 or D_3. - N. J. A. Sloane, Jan 11 2016
For n > 2 the generalized Pell's equation x^2 - 2*(a(n) - 2)y^2 = (a(n) - 4)^2 has a finite number of positive integer solutions. - Muniru A Asiru, Apr 19 2016
Union of A188896, A277449, {1,4}. - Muniru A Asiru, Nov 25 2016
Interleaving of A008527 and A108099. - Bruce J. Nicholson, Oct 14 2019

			G.f. = 1 + 4*x + 10*x^2 + 20*x^3 + 34*x^4 + 52*x^5 + 74*x^6 + 100*x^7 + ...

N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10b, page 231, W_a(t).
H. S. M. Coxeter, "Polyhedral numbers," in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #28.
R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.
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
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
J. M. Grau, C. Miguel, and A. M. Oller-Marcén, Generalized Quaternion Rings over Z/nZ for an odd n, arXiv:1706.04760 [math.RA], 2017. See Theorem 1, p. 10.
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy.]
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010), Article 10.7.8.
M. O'Keeffe, N-dimensional diamond, sodalite and rare sphere packings, Acta Cryst., A 47 (1991), 749-753.
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, sod.
Aditya Sivakumar and Dmitri Tymoczko, Intuitive Musical Homotopy, 2018.
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558. DOI: 10.1021/ic00220a025.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000292, A053545, A206399.
Cf. similar sequences listed in A255843.
The growth series for the affine Coxeter groups D_3 through D_12 are A005893 and A266759-A266767.
For partial sums see A005894.
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.

[2*n^2-0^n+2: n in [0..60]]; // Vincenzo Librandi, Sep 27 2011

A005893:=-(z+1)*(1+z^2)/(z-1)^3; # Simon Plouffe in his 1992 dissertation

Join[{1}, Table[2*(n + 1)^2 + 2, {n, 0, 200}]] (* Vladimir Joseph Stephan Orlovsky, Jul 10 2011 *)
Join[{1},LinearRecurrence[{3,-3,1},{4,10,20},50]] (* Harvey P. Dale, Feb 26 2012 *)
a[ n_] := SeriesCoefficient[ (1 - x^4) / (1 - x)^4, {x, 0, Abs@n}]; (* Michael Somos, May 14 2014 *)
a[ n_] := 2 n^2 + 2 - Boole[n == 0]; (* Michael Somos, May 14 2014 *)

a(n)=2*n^2-0^n+2 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: (1 - x^4)/(1-x)^4.
a(n) = A071619(n-1) + A071619(n) + A071619(n+1), n > 0. - Ralf Stephan, Apr 26 2003
a(n) = binomial(n+3, 3) - binomial(n-1, 3) for n >= 1. - Mitch Harris, Jan 08 2008
a(n) = (n+1)^2 + (n-1)^2. - Benjamin Abramowitz, Apr 14 2009
a(n) = A000217(n-2) + A000217(n-1) + A000217(n) + A000217(n+1) for n >= 2. - Rick L. Shepherd, Sep 30 2009
a(n) = 2*n^2 - 0^n + 2. - Vincenzo Librandi, Sep 27 2011
a(0)=1, a(1)=4, a(2)=10, a(3)=20, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 26 2012
a(n) = A228643(n+1,2) for n > 0. - Reinhard Zumkeller, Aug 29 2013
a(n) = a(-n) for all n in Z. - Michael Somos, May 14 2014
For n >= 2: a(n) = a(n-1) + 4*n - 2. - Bob Selcoe, Mar 22 2016
E.g.f.: -1 + 2*(1 + x + x^2)*exp(x). - Ilya Gutkovskiy, Apr 19 2016
a(n) = 2*A002522(n), n>0. - R. J. Mathar, May 30 2022
From Amiram Eldar, Sep 16 2022: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi)*Pi + 3)/4.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi)*Pi + 3)/4. (End)
Empirical: Integral_{u=-oo..+oo} sigmoid(u)*log(sigmoid(n * u)) du = -Pi^2*a(n) / (24*n), where sigmoid(x) = 1/(1+exp(-x)). Also works for non-integer n>0. - Carlo Wood, Dec 04 2023
Let P(k,n) be the n-th k-gonal number. Then P(a(k),n) = (k*n-k+1)^2 + (k-1)^2*(n-1). - Charlie Marion, May 15 2024

A005899 Number of points on surface of octahedron; also coordination sequence for cubic lattice: a(0) = 1; for n > 0, a(n) = 4n^2 + 2.

Original entry on oeis.org

1, 6, 18, 38, 66, 102, 146, 198, 258, 326, 402, 486, 578, 678, 786, 902, 1026, 1158, 1298, 1446, 1602, 1766, 1938, 2118, 2306, 2502, 2706, 2918, 3138, 3366, 3602, 3846, 4098, 4358, 4626, 4902, 5186, 5478, 5778, 6086, 6402, 6726, 7058, 7398, 7746, 8102, 8466

Offset: 0

N. J. A. Sloane

Also, the number of regions the plane can be cut into by two overlapping concave (2n)-gons. - Joshua Zucker, Nov 05 2002
If X is an n-set and Y_i (i=1,2,3) are mutually disjoint 2-subsets of X then a(n-5) is equal to the number of 5-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Aug 26 2007
Binomial transform of a(n) is A055580(n). - Wesley Ivan Hurt, Apr 15 2014
The identity (4*n^2+2)^2 - (n^2+1)*(4*n)^2 = 4 can be written as a(n)^2 - A002522(n)*A008586(n)^2 = 4. - Vincenzo Librandi, Jun 15 2014
Also the least number of unit cubes required, at the n-th iteration, to surround a 3D solid built from unit cubes, in order to hide all its visible faces, starting with a unit cube. - R. J. Cano, Sep 29 2015
Also, coordination sequence for "tfs" 3D uniform tiling. - N. J. A. Sloane, Feb 10 2018
Also, the number of n-th order specular reflections arriving at a receiver point from an emitter point inside a cuboid with reflective faces. - Michael Schutte, Sep 18 2018

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) cF8
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tilings #16 and #22.
R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.
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
Barry Balof, Restricted tilings and bijections, J. Integer Seq. 15 (2012), no. 2, Article 12.2.3, 17 pp.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Pierre de la Harpe, On the prehistory of growth of groups, arXiv:2106.02499 [math.GR], 2021.
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.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
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]
Carlos I. Perez-Sanchez, The Spectral Action on quivers, arXiv:2401.03705 [math.RT], 2024.
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 pcu tiling (or net)
Reticular Chemistry Structure Resource (RCSR), The tfs tiling (or net)
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558.
N. J. A. Sloane, Illustration of a(0)=1, a(1)=6, a(2)=18 (from Teo-Sloane 1985)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Partial sums give A001845.
Column 2 * 2 of array A188645.
Cf. A001844, A002522, A008586, A010014, A055580, A195322, A206399.
Cf. A000384, A014105, A000217, A008574, A008412, A035597.
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.
Row 3 of A035607, A266213, A343599.
Column 3 of A113413, A119800, A122542.

[4*n^2 + 2 : n in [0..50]]; // Wesley Ivan Hurt, Oct 26 2015

A005899:=n->4*n^2 + 2; seq(A005899(n), n=0..50); # Wesley Ivan Hurt, Apr 15 2014

Join[{1},4Range[40]^2+2] (* or *) Join[{1},LinearRecurrence[{3,-3,1},{6,18,38},40]] (* Harvey P. Dale, Nov 08 2011 *)

Vec(((1+x)/(1-x))^3 + O(x^100)) \\ Altug Alkan, Oct 26 2015

G.f.: ((1+x)/(1-x))^3. - Simon Plouffe in his 1992 dissertation
Binomial transform of [1, 5, 7, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Nov 02 2007
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=1, a(1)=6, a(2)=18, a(3)=38. - Harvey P. Dale, Nov 08 2011
Recurrence: n*a(n) = (n-2)*a(n-2) + 6*a(n-1), a(0)=1, a(1)=6. - Fung Lam, Apr 15 2014
For n > 0, a(n) = A001844(n-1) + A001844(n) = (n-1)^2 + 2n^2 + (n+1)^2. - Doug Bell, Aug 18 2015
For n > 0, a(n) = A010014(n) - A195322(n). - R. J. Cano, Sep 29 2015
For n > 0, a(n) = A000384(n+1) + A014105(n-1). - Bruce J. Nicholson, Oct 08 2017
a(n) = A008574(n) + A008574(n-1) + a(n-1). - Bruce J. Nicholson, Dec 18 2017
a(n) = 2*d*Hypergeometric2F1(1-d, 1-n, 2, 2) where d=3, n>0. - Shel Kaphan, Feb 16 2023
a(n) = A035597(n)*3/n, for n>0. - Shel Kaphan, Feb 26 2023
E.g.f.: exp(x)*(2 + 4*x + 4*x^2) - 1. - Stefano Spezia, Mar 08 2023
Sum_{n>=0} 1/a(n) = 3/4 + Pi *sqrt(2)*coth( Pi/sqrt 2)/8 = 1.31858... - R. J. Mathar, Apr 27 2024

A005894 Centered tetrahedral numbers.

Original entry on oeis.org

1, 5, 15, 35, 69, 121, 195, 295, 425, 589, 791, 1035, 1325, 1665, 2059, 2511, 3025, 3605, 4255, 4979, 5781, 6665, 7635, 8695, 9849, 11101, 12455, 13915, 15485, 17169, 18971, 20895, 22945, 25125, 27439, 29891, 32485, 35225, 38115

Offset: 0

N. J. A. Sloane

Binomial transform of (1,4,6,4,0,0,0,...). - Paul Barry, Jul 01 2003

If X is an n-set and Y a fixed 4-subset of X then a(n-4) is equal to the number of 4-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007

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
Milan Janjic, Two Enumerative Functions
T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (10).
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
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

(1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.
Cf. A000292.
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.

[(2*n+1)*(n^2+n+3)/3: n in [0..30]]; // G. C. Greubel, Nov 30 2017

Table[(2n+1)(n^2+n+3)/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,5,15,35},40] (* Harvey P. Dale, Nov 03 2011 *)

a(n)=(2*n+1)*(n^2+n+3)/3 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (2*n + 1)*(n^2 + n + 3)/3.
G.f.: (1+x)*(1+x^2)/(1-x)^4. - Simon Plouffe in his 1992 dissertation
a(n) = C(n, 0) + 4*C(n, 1) + 6*C(n, 2) + 4*C(n, 3). - Paul Barry, Jul 01 2003
a(n) is the sum of 4 consecutive tetrahedral (or pyramidal) numbers: a(n) = A000292(n-3) + A000292(n-2) + A000292(n-1) + A000292(n). - Alexander Adamchuk, May 20 2006
a(n) = binomial(n+3,n) + binomial(n+2,n-1) + binomial(n+1,n-2) + binomial(n,n-3). (modified by G. C. Greubel, Nov 30 2017)
a(n) = a(n-1) + 2*n^2 + 2, n>=1 (first differences A005893). - Vincenzo Librandi, Mar 27 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=5, a(2)=15, a(3)=35. - Harvey P. Dale, Nov 03 2011
E.g.f.: (3 + 12*x + 9*x^2 + 2*x^3)*exp(x)/3. - G. C. Greubel, Nov 30 2017
a(n) = A006527(n)+A006527(n+1) = A000330(n-1)+A000330(n+1). - R. J. Mathar, Jun 05 2025

A008137 Coordination sequence T1 for Zeolite Code LTA and RHO.

Original entry on oeis.org

1, 4, 9, 17, 28, 42, 60, 81, 105, 132, 162, 196, 233, 273, 316, 362, 412, 465, 521, 580, 642, 708, 777, 849, 924, 1002, 1084, 1169, 1257, 1348, 1442, 1540, 1641, 1745, 1852, 1962, 2076, 2193, 2313, 2436, 2562, 2692, 2825, 2961, 3100, 3242, 3388, 3537, 3689

Offset: 0

Ralf W. Grosse-Kunstleve

Also, growth series for the affine Coxeter (or Weyl) groups B_3. - N. J. A. Sloane, Jan 11 2016

Also, coordination sequence for "rho" 3D uniform tiling. - N. J. A. Sloane, Feb 10 2018

N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10b, page 231, W_a(t).
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tilings #25 and 27.
W. M. Meier, D. H. Olson and Ch. Baerlocher, Atlas of Zeolite Structure Types, 4th Ed., Elsevier, 1996.

R. W. Grosse-Kunstleve, 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 (RCSR), The lta tiling (or net)
Reticular Chemistry Structure Resource (RCSR), The rho tiling (or net)
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

The growth series for the finite Coxeter (or Weyl) groups B_2 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858. These are all rows of A128084. The growth series for the affine Coxeter (or Weyl) groups B_2 through B_12 are A008576, A008137, A267167-A267175.
For partial sums see A299276.
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.

(1-x^2)*(1-x^4)*(1-x^6)/((1-x)^4*(1-x^3)*(1-x^5));
seq(coeff(series(%,x,n+1),x,n), n=0..48);

a(5*m+k) = 40*m^2 + 16*k*m + one of 5 numbers depending on k, 0 <= k < 5 (N. J. A. Sloane).
G.f.: (1-x^2)*(1-x^4)*(1-x^6)/((1-x)^4*(1-x^3)*(1-x^5)). This can also be written as (x+1)^3*(x^2+1)*(x^2-x+1)/((1-x)^3*(x^4+x^3+x^2+x+1)). - N. J. A. Sloane, Feb 10 2018
a(n) = 12/5 - 0^n + (8/5)*n^2 - (1/25)*(5+sqrt(5))*cos(2*Pi*n/5) - (1/25)*(5-sqrt(5))*cos(4*Pi*n/5). - Eric Simon Jacob, Feb 12 2023

A063489 a(n) = (2n-1)(5n^2-5n+6)/6.

Original entry on oeis.org

1, 8, 30, 77, 159, 286, 468, 715, 1037, 1444, 1946, 2553, 3275, 4122, 5104, 6231, 7513, 8960, 10582, 12389, 14391, 16598, 19020, 21667, 24549, 27676, 31058, 34705, 38627, 42834, 47336, 52143, 57265, 62712, 68494, 74621, 81103, 87950, 95172, 102779, 110781, 119188

Offset: 1

N. J. A. Sloane, Aug 01 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000
T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (10).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.
Partial sums of A010001.
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.

[(2*n-1)*(5*n^2-5*n+6)/6: n in [1..30]]; // G. C. Greubel, Dec 01 2017

Table[(2n-1)(5n^2-5n+6)/6,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,8,30,77},40] (* Harvey P. Dale, Aug 20 2012 *)

a(n) = { (2*n - 1)*(5*n^2 - 5*n + 6)/6 } \\ Harry J. Smith, Aug 23 2009

my(x='x+O('x^30)); Vec(serlaplace((-6 + 12*x + 15*x^2 + 10*x^3 )*exp(x)/6 + 1)) \\ G. C. Greubel, Dec 01 2017

G.f.: x*(1+x)*(1+3*x+x^2)/(1-x)^4. - Colin Barker, Mar 02 2012
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4), with a(1)=1, a(2)=8, a(3)=30, a(4)=77. - Harvey P. Dale, Aug 20 2012
E.g.f.: (-6 + 12*x + 15*x^2 + 10*x^3)*exp(x)/6 + 1. - G. C. Greubel, Dec 01 2017

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.

Original entry on oeis.org

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

A005917 Rhombic dodecahedral numbers: a(n) = n^4 - (n - 1)^4.

Original entry on oeis.org

1, 15, 65, 175, 369, 671, 1105, 1695, 2465, 3439, 4641, 6095, 7825, 9855, 12209, 14911, 17985, 21455, 25345, 29679, 34481, 39775, 45585, 51935, 58849, 66351, 74465, 83215, 92625, 102719, 113521, 125055, 137345, 150415, 164289, 178991

Offset: 1

N. J. A. Sloane

Final digits of a(n), i.e., a(n) mod 10, are repeated periodically with period of length 5 {1,5,5,5,9}. There is a symmetry in this list since the sum of two numbers equally distant from the ends is equal to 10 = 1 + 9 = 5 + 5 = 2*5. Last two digits of a(n), i.e., a(n) mod 100, are repeated periodically with period of length 50. - Alexander Adamchuk, Aug 11 2006
a(n) = VarScheme(n,2) in the scheme displayed in A128195. - Peter Luschny, Feb 26 2007
If Y is a 3-subset of a 2n-set X then, for n >= 2, a(n-2) is the number of 4-subsets of X intersecting Y. - Milan Janjic, Nov 18 2007
The numbers are the constant number found in magic squares of order n, where n is an odd number, see the comment in A006003. A Magic Square of side 1 is 1; 3 is 15; 5 is 65 and so on. - David Quentin Dauthier, Nov 07 2008
Two times the area of the triangle with vertices at (0,0), ((n - 1)^2, n^2), and (n^2, (n - 1)^2). - J. M. Bergot, Jun 25 2013
Bisection of A006003. - Omar E. Pol, Sep 01 2018
Construct an array M with M(0,n) = 2*n^2 + 4*n + 1 = A056220(n+1), M(n,0) = 2*n^2 + 1 = A058331(n) and M(n,n) = 2*n*(n+1) + 1 = A001844(n). Row(n) begins with all the increasing odd numbers from A058331(n) to A001844(n) and column(n) begins with all the decreasing odd numbers from A056220(n+1) to A001844(n). The sum of the terms in row(n) plus those in column(n) minus M(n,n) equals a(n+1). The first five rows of array M are [1, 7, 17, 31, 49, ...]; [3, 5, 15, 29, 47, ...]; [9, 11, 13, 27, 45, ...]; [19, 21, 23, 25, 43, ...]; [33, 35, 37, 39, 41, ...]. - J. M. Bergot, Jul 16 2013 [This contribution was moved here from A047926 by Petros Hadjicostas, Mar 08 2021.]
For n>=2, these are the primitive sides s of squares of type 2 described in A344332. - Bernard Schott, Jun 04 2021
(a(n) + 1) / 2 = A212133(n) is the number of cells in the n-th rhombic-dodecahedral polycube. - George Sicherman, Jan 21 2024

J. H. Conway and R. K. Guy, The Book of Numbers, p. 53.
E. Deza and M. M. Deza, Figurate Numbers, World Scientific Publishing, 2012, pp. 123-124.
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 = 1..10000
Mario Defranco and Paul E. Gunnells, Hypergraph matrix models and generating functions, arXiv:2204.11361 [math.CO], 2022.
Milan Janjic, Two Enumerative Functions
T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (9).
Andy Nicol, Illustration of Rhombic Dodecahedral Numbers
C. J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq. 16 (2013) #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
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Eric Weisstein's World of Mathematics, Rhombic Dodecahedral Number.
Eric Weisstein's World of Mathematics, Nexus Number.
D. Zeitlin, A family of Galileo sequences, Amer. Math. Monthly 82 (1975), 819-822.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

(1/12)*t*(2*n^3 - 3*n^2 + n) + 2*n - 1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A063493, A063494, A063495, A063496.
Column k=3 of A047969.
Cf. A128195, A176850, A005408, A176271, A212133.
Cf. A001844, A000583, A000290.
Cf. A007588, A000292, A000332, A002817, A342354.
Cf. A031215, A008292.
Cf. A016754, A344330, A344332.

a005917 n = a005917_list !! (n-1)
a005917_list = map sum $ f 1 [1, 3 ..] where
   f x ws = us : f (x + 2) vs where (us, vs) = splitAt x ws
-- Reinhard Zumkeller, Nov 13 2014

[n^4 - (n-1)^4: n in [1..50]]; // Vincenzo Librandi, Aug 01 2011

Table[n^4-(n-1)^4,{n,40}]  (* Harvey P. Dale, Apr 01 2011 *)
#[[2]]-#[[1]]&/@Partition[Range[0,40]^4,2,1] (* More efficient than the above Mathematica program because it only has to calculate each 4th power once *) (* Harvey P. Dale, Feb 07 2015 *)
Differences[Range[0,40]^4] (* Harvey P. Dale, Aug 11 2023 *)

a(n)=n^4-(n-1)^4 \\ Charles R Greathouse IV, Jul 31 2011

A005917_list, m = [], [24, -12, 2, 1]
for _ in range(10**2):
    A005917_list.append(m[-1])
    for i in range(3):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

a(n) = (2*n - 1)*(2*n^2 - 2*n + 1).
Sum_{i=1..n} a(i) = n^4 = A000583(n). First differences of A000583.
G.f.: x*(1+x)*(1+10*x+x^2)/(1-x)^4. - Simon Plouffe in his 1992 dissertation
More generally, g.f. for n^m - (n - 1)^m is Euler(m, x)/(1 - x)^m, where Euler(m, x) is Eulerian polynomial of degree m (cf. A008292). E.g.f.: x*(exp(y/(1 - x)) - exp(x*y/(1 - x)))/(exp(x*y/(1 - x))-x*exp(y/(1 - x))). - Vladeta Jovovic, May 08 2002
a(n) = sum of the next (2*n - 1) odd numbers; i.e., group the odd numbers so that the n-th group contains (2*n - 1) elements like this: (1), (3, 5, 7), (9, 11, 13, 15, 17), (19, 21, 23, 25, 27, 29, 31), ... E.g., a(3) = 65 because 9 + 11 + 13 + 15 + 17 = 65. - Xavier Acloque, Oct 11 2003
a(n) = 2*n - 1 + 12*Sum_{i = 1..n} (i - 1)^2. - Xavier Acloque, Oct 16 2003
a(n) = (4*binomial(n,2) + 1)*sqrt(8*binomial(n,2) + 1). - Paul Barry, Mar 14 2004
Binomial transform of [1, 14, 36, 24, 0, 0, 0, ...], if the offset is 0. - Gary W. Adamson, Dec 20 2007
Sum_{i=1..n-1}(a(i) + a(i+1)) = 8*Sum_{i=1..n}(i^3 + i) = 16*A002817(n-1) for n > 1. - Bruno Berselli, Mar 04 2011
a(n+1) = a(n) + 2*(6*n^2 + 1) = a(n) + A005914(n). - Vincenzo Librandi, Mar 16 2011
a(n) = -a(-n+1). a(n) = (1/6)*(A181475(n) - A181475(n-2)). - Bruno Berselli, Sep 26 2011
a(n) = A045975(2*n-1,n) = A204558(2*n-1)/(2*n - 1). - Reinhard Zumkeller, Jan 18 2012
a(n+1) = Sum_{k=0..2*n+1} (A176850(n,k) - A176850(n-1,k))*(2*k + 1), n >= 1. - L. Edson Jeffery, Nov 02 2012
a(n) = A005408(n-1) * A001844(n-1) = (2*(n - 1) + 1) * (2*(n - 1)*n + 1) = A000290(n-1)*12 + 2 + a(n-1). - Bruce J. Nicholson, May 17 2017
a(n) = A007588(n) + A007588(n-1) = A000292(2n-1) + A000292(2n-2) + A000292(2n-3) = A002817(2n-1) - A002817(2n-2). - Bruce J. Nicholson, Oct 22 2017
a(n) = A005898(n-1) + 6*A000330(n-1) (cf. Deza, Deza, 2012, p. 123, Section 2.6.2). - Felix Fröhlich, Oct 01 2018
a(n) = A300758(n-1) + A005408(n-1). - Bruce J. Nicholson, Apr 23 2020
G.f.: polylog(-4, x)*(1-x)/x. See the Simon Plouffe formula above (with expanded numerator), and the g.f. of the rows of A008292 by Vladeta Jovovic, Sep 02 2002. - Wolfdieter Lang, May 10 2021

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.

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

cp's OEIS Frontend

A005893 Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A005899 Number of points on surface of octahedron; also coordination sequence for cubic lattice: a(0) = 1; for n > 0, a(n) = 4n^2 + 2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A005894 Centered tetrahedral numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A008137 Coordination sequence T1 for Zeolite Code LTA and RHO.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Formula

A063489 a(n) = (2*n-1)*(5*n^2-5*n+6)/6.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

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.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A063489 a(n) = (2n-1)(5n^2-5n+6)/6.