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A005899 Number of points on surface of octahedron: a(0) = 1; for n>0, ( or ) a(n) = 4n^2 + 2, coordination sequence for cubic lattice.
(Formerly M4115)
18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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) cF8

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).

LINKS

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).

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

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.

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 (3, -3, 1).

FORMULA

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(0)=1, a(1)=6, a(2)=18, a(3)=38, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - 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

a(0)=1; 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

MAPLE

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

MATHEMATICA

s=2; lst={s-1}; Do[s+=n+1; AppendTo[lst, s], {n, 3, 6!, 8}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)

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

PROG

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

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

CROSSREFS

Partial sums give A001845.

Column 2 * 2 of array A188645.

Cf. A001844, A002522, A008586, A010014, A055580, A195322, A206399.

Sequence in context: A185223 A101853 A132432 * A261652 A180118 A270335

Adjacent sequences:  A005896 A005897 A005898 * A005900 A005901 A005902

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 20 17:47 EST 2017. Contains 282392 sequences.