%I M4497 #121 Apr 27 2024 14:34:01

%S 1,8,26,56,98,152,218,296,386,488,602,728,866,1016,1178,1352,1538,

%T 1736,1946,2168,2402,2648,2906,3176,3458,3752,4058,4376,4706,5048,

%U 5402,5768,6146,6536,6938,7352,7778,8216,8666,9128,9602,10088,10586

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

%C 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).

%C Coordination sequence for b.c.c. lattice.

%C Also coordination sequence for 3D uniform tiling with tile an equilateral triangular prism. - _N. J. A. Sloane_, Feb 06 2018

%C Binomial transform of [1, 7, 11, 1, -1, 1, -1, 1, ...]. - _Gary W. Adamson_, Oct 22 2007

%C First differences of A005898. - _Jonathan Vos Post_, Feb 06 2011

%C 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

%C 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

%C Number of cubes of edge length 1 required to make a hollow cube of edge length n+1. - _Peter M. Chema_, Apr 01 2017

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

%D Gmelin Handbook of Inorg. and Organomet. Chem., 8th Ed., 1994, TYPIX search code (194) hP4

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

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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A005897/b005897.txt">Table of n, a(n) for n = 0..10000</a>

%H John Elias, <a href="/A005897/a005897.png">Illustration: Generalized octagonal cubes</a>

%H R. W. Grosse-Kunstleve, <a href="/A005897/a005897.html">Coordination Sequences and Encyclopedia of Integer Sequences</a>

%H R. W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, <a href="http://neilsloane.com/doc/ac96cs/">Algebraic Description of Coordination Sequences and Exact Topological Densities for Zeolites</a>, Acta Cryst., A52 (1996), pp. <a href="http://dx.doi.org/10.1107/S0108767396007519">879-889</a>.

%H M. O'Keeffe, <a href="http://dx.doi.org/10.1524/zkri.1995.210.12.905">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908.

%H M. O'Keeffe, <a href="/A008527/a008527.pdf">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/nets/hex">The hex tiling (or net)</a>

%H B. K. Teo and N. J. A. Sloane, <a href="http://dx.doi.org/10.1021/ic00220a025">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985), 4545-4558.

%H <a href="/index/Ba#bcc">Index entries for sequences related to b.c.c. lattice</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: (1+x)*(1+4*x+x^2)/(1-x)^3. - _Simon Plouffe_

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

%F 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

%F a(n) = A033581(n) + 2. - _Reinhard Zumkeller_, Apr 27 2014

%F E.g.f.: 2*(1 + 3*x + 3*x^2)*exp(x) - 1. - _G. C. Greubel_, Dec 01 2017

%F a(n) = A000567(n+1) + A045944(n-1), for n>0. See illustration. - _John Elias_, Mar 12 2022

%F a(n) = 2*A056107(n), n>0. - _R. J. Mathar_, May 30 2022

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

%e 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.

%p A005897:=-(z+1)*(z**2+4*z+1)/(z-1)**3; # conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation

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

%o (Magma) [1] cat [6*n^2 + 2: n in [1..50]]; // _Vincenzo Librandi_, Oct 26 2011

%o (PARI) a(n)=if(n,6*n^2+2,1) \\ _Charles R Greathouse IV_, Mar 06 2014

%o (PARI) x='x+O('x^30); Vec(serlaplace(2*(1 + 3*x + 3*x^2)*exp(x) - 1)) \\ _G. C. Greubel_, Dec 01 2017

%o (Haskell) a005897 n = if n == 0 then 1 else 6 * n ^ 2 + 2 -- _Reinhard Zumkeller_, Apr 27 2014

%Y Cf. A000578, A206399.

%Y See A005898 for partial sums.

%Y 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.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_, _Ralf W. Grosse-Kunstleve_