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

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

Author

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

Keywords

• nonn
• easy
• nice

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

a(n) is the number of pieces on the outside of a (n+1) X (n+1) X (n+1) Rubik's cube. For n > 0: corners = 8, edges = 12*(n-1), center pieces = 6*(n-1)^2. - Demilade Runsewe, Jan 08 2025

Examples

			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.
		

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).

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.

Programs

Haskell

a005897 n = if n == 0 then 1 else 6 * n ^ 2 + 2 -- Reinhard Zumkeller, Apr 27 2014

Magma

[1] cat [6*n^2 + 2: n in [1..50]]; // Vincenzo Librandi, Oct 26 2011

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 *)

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

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

a(n) = 8 + 12*(n-1) + 6*(n-1)^2 for n > 0. - Demilade Runsewe, Jan 08 2025