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. (Formerly M4834)

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, 16812, 17642, 18492, 19362

Offset

0,2

Comments

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

References

H. S. M. Coxeter, "Polyhedral numbers," in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 134.

Gmelin Handbook of Inorg. and Organomet. Chem., 8th Ed., 1994, TYPIX search code (225) cF4

Robert W. Marks and R. Buckminster Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.

Sidney 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).

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Michael Baake, Uwe Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.

Roland Bacher, Pierre de la Harpe, and Boris 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).

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.

Branko Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #1.

Gabriele 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.

Boon K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

Kirby Urner, Microarchitecture of the Virus.

R. Vaughan and 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).

Formula

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

Sum_{n>=0} (-1)^n/a(n) = 3/4 - Pi*sqrt(5)*cosech(Pi/sqrt(5))/20. - Amiram Eldar, Nov 01 2025

Mathematica

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

Prog

(PARI) a(n)=if(n<0, 0, 10*n^2+1+(n>0))
(Magma) [n eq 0 select 1 else 2*(5*n^2+1): n in [0..55]]; // G. C. Greubel, May 25 2023
(SageMath) [2*(5*n^2 + 1)-int(n==0) for n in range(56)] # G. C. Greubel, May 25 2023

Crossrefs

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.

Sequence in context: A282693 A045945 A210206 * A090554 A009948 A193068

Adjacent sequences: A005898 A005899 A005900 * A005902 A005903 A005904

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, R. Vaughan

Status

approved