login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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)
13
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 (list; graph; refs; listen; history; text; internal format)
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.

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

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).

LINKS

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

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

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

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).

FORMULA

G.f. for coordination sequence for A_n lattice is Sum(binomial(n, i)^2*z^i, i=0..n)/(1-z)^n. [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

MAPLE

A005901:=-(z+1)*(z**2+8*z+1)/(z-1)**3; # Simon Plouffe in his 1992 dissertation

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))

CROSSREFS

Cf. A004015, A206399.

Sequence in context: A282693 A045945 A210206 * A090554 A009948 A193068

Adjacent sequences:  A005898 A005899 A005900 * A005902 A005903 A005904

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane, R. Vaughan

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified October 21 07:48 EDT 2017. Contains 293679 sequences.