A005897 a(n) = 6*n^2 + 2 for n > 0, a(0)=1. (Formerly M4497)

1, 8, 26, 56, 98, 152, 218, 296, 386, 488, 602, 728, 866, 1016, 1178, 1352, 1538, 1736, 1946, 2168, 2402, 2648, 2906, 3176, 3458, 3752, 4058, 4376, 4706, 5048, 5402, 5768, 6146, 6536, 6938, 7352, 7778, 8216, 8666, 9128, 9602, 10088, 10586

Offset

0,2

Comments

Number of points on surface of 3-dimensional cube in which each face has a square grid of dots drawn on it (with n+1 points along each edge, including the corners).

Coordination sequence for b.c.c. lattice.

Also coordination sequence for 3D uniform tiling with tile an equilateral triangular prism. - N. J. A. Sloane, Feb 06 2018

Binomial transform of [1, 7, 11, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Oct 22 2007

First differences of A005898. - Jonathan Vos Post, Feb 06 2011

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=2, s=1. After 8, all terms are in A000408. - Bruno Berselli, Feb 07 2012

For n > 0, the sequence of last digits (i.e., a(n) mod 10) is (8, 6, 6, 8, 2) repeating forever. - M. F. Hasler, Apr 05 2016

Number of cubes of edge length 1 required to make a hollow cube of edge length n+1. - Peter M. Chema, Apr 01 2017

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 (194) hP4

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #11.

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

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

John Elias, Illustration: Generalized octagonal cubes

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.

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 hex tiling (or net)

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

Index entries for sequences related to b.c.c. lattice

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

G.f.: (1+x)*(1+4*x+x^2)/(1-x)^3. - Simon Plouffe

a(0) = 1, a(n) = (n+1)^3 - (n-1)^3. - Ilya Nikulshin (ilyanik(AT)gmail.com), Aug 11 2009

a(0)=1, a(1)=8, a(2)=26, a(3)=56; for n>3, a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Oct 25 2011

a(n) = A033581(n) + 2. - Reinhard Zumkeller, Apr 27 2014

E.g.f.: 2*(1 + 3*x + 3*x^2)*exp(x) - 1. - G. C. Greubel, Dec 01 2017

a(n) = A000567(n+1) + A045944(n-1), for n>0. See illustration. - John Elias, Mar 12 2022

a(n) = 2*A056107(n), n>0. - R. J. Mathar, May 30 2022

Sum_{n>=0} 1/a(n) = 3/4+ Pi*sqrt(3)*coth(Pi/sqrt 3)/12 = 1.2282133.. - R. J. Mathar, Apr 27 2024

Example

For n = 1 we get the 8 corners of the cube; for n = 2 each face has 9 points, for a total of 8 + 12 + 6 = 26.

Maple

A005897:=-(z+1)*(z**2+4*z+1)/(z-1)**3; # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

Mathematica

Join[{1}, 6Range[50]^2+2] (* or *) Join[{1}, LinearRecurrence[{3, -3, 1}, {8, 26, 56}, 50]] (* Harvey P. Dale, Oct 25 2011 *)

Prog

(Magma) [1] cat [6*n^2 + 2: n in [1..50]]; // Vincenzo Librandi, Oct 26 2011
(PARI) a(n)=if(n, 6*n^2+2, 1) \\ Charles R Greathouse IV, Mar 06 2014
(PARI) x='x+O('x^30); Vec(serlaplace(2*(1 + 3*x + 3*x^2)*exp(x) - 1)) \\ G. C. Greubel, Dec 01 2017
(Haskell) a005897 n = if n == 0 then 1 else 6 * n ^ 2 + 2 -- Reinhard Zumkeller, Apr 27 2014

Crossrefs

Cf. A000578, A206399.

See A005898 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.

Sequence in context: A347677 A225274 A085690 * A215097 A331242 A111694

Adjacent sequences: A005894 A005895 A005896 * A005898 A005899 A005900

Keyword

nonn,easy,nice,changed

Author

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

Status

approved