A005902 - OEIS

%I M4898 #105 Sep 08 2022 08:44:34

%S 1,13,55,147,309,561,923,1415,2057,2869,3871,5083,6525,8217,10179,

%T 12431,14993,17885,21127,24739,28741,33153,37995,43287,49049,55301,

%U 62063,69355,77197,85609,94611,104223,114465,125357,136919,149171,162133,175825,190267,205479

%N Centered icosahedral (or cuboctahedral) numbers, also crystal ball sequence for f.c.c. lattice.

%C Called "magic numbers" in some chemical contexts.

%C Partial sums of A005901(n). - _Lekraj Beedassy_, Oct 30 2003

%C Equals binomial transform of [1, 12, 30, 20, 0, 0, 0, ...]. - _Gary W. Adamson_, Aug 01 2008

%C Crystal ball sequence for A_3 lattice. - _Michael Somos_, Jun 03 2012

%D H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.

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

%H T. D. Noe, <a href="/A005902/b005902.txt">Table of n, a(n) for n = 0..1000</a>

%H S. Bjornholm, <a href="http://dx.doi.org/10.1080/00107519008213781">Clusters, condensed matter in embryonic form</a>, Contemp. Phys. 31 1990 pp. 309-324.

%H J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).

%H Nicolas Gastineau, Olivier Togni, <a href="https://arxiv.org/abs/1806.08136">Coloring of the d-th power of the face-centered cubic grid</a>, arXiv:1806.08136 [cs.DM], 2018.

%H D. R. Herrick, <a href="https://chemistry.uoregon.edu/profile/dherrick/">Home Page</a> (displays these numbers as sizes of clusters in chemistry)

%H Xiaogang Liang, Ilyar Hamid, and Haiming Duan, <a href="https://doi.org/10.1063/1.4954741">Dynamic stabilities of icosahedral-like clusters and their ability to form quasicrystals</a>,>, AIP Advances 6, 065017 (2016).

%H T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Reports, 273 (1996), 199-241, eq. (11).

%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 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 K. Urner, <a href="http://www.4dsolutions.net/ocn/sphpack2.html">Cuboctahedral Sphere Packing</a>

%H <a href="/index/Cor#crystal_ball">Index entries for crystal ball sequences</a>

%H <a href="/index/Fa#fcc">Index entries for sequences related to f.c.c. lattice</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = (2*n+1)*(5*n^2+5*n+3)/3.

%F For n > 0, n*a(n) = (Sum_{i=0..n-1} a(i)) + 2*A005891(n)*A000217(n). - _Bruno Berselli_, Feb 02 2011

%F a(-1 - n) = -a(n). - _Michael Somos_, Jun 03 2012

%F From _Indranil Ghosh_, Apr 08 2017: (Start)

%F G.f.: (x^3 + 9x^2 + 9x + 1)/(x - 1)^4.

%F E.g.f.: (1/3)*exp(x)*(10x^3 + 45x^2 + 36x + 3).

%F (End)

%F a(n) = A100171(n+1) - A008778(n-1) = A100174(n+1) - A000290(n) = A005917(n+1) - A006331(n) = A051673(n+1) + A000578(n). - _Bruce J. Nicholson_, Jul 05 2018

%e a(4) = 147 = (1, 3, 3, 1) dot (1, 12, 30, 20) = (1 + 36 + 90 + 20). - _Gary W. Adamson_, Aug 01 2008

%e G.f. = 1 + 13*x + 55*x^2 + 147*x^3 + 309*x^4 + 561*x^5 + 923*x^6 + 1415*x^7 + ...

%p A005902 := n -> (2*n+1)*(5*n^2+5*n+3)/3;

%p A005902:=(z+1)*(z**2+8*z+1)/(z-1)**4; # _Simon Plouffe_ in his 1992 dissertation

%t f[n_] := (2n + 1)(5n^2 + 5n + 3)/3; Array[f, 36, 0] (* _Robert G. Wilson v_, Feb 02 2011 *)

%t LinearRecurrence[{4,-6,4,-1},{1,13,55,147},50] (* _Harvey P. Dale_, Oct 08 2015 *)

%t CoefficientList[Series[(x^3 + 9*x^2 + 9*x + 1)/(x - 1)^4, {x, 0, 50}], x] (* _Indranil Ghosh_, Apr 08 2017 *)

%o (PARI) {a(n) = (2*n + 1) * (5*n^2 + 5*n + 3) / 3}; /* _Michael Somos_, Jun 03 2012 */

%o (PARI) x='x+O('x^50); Vec((x^3 + 9*x^2 + 9*x + 1)/(x - 1)^4) \\ _Indranil Ghosh_, Apr 08 2017

%o (Magma) [(2*n+1)*(5*n^2+5*n+3)/3: n in [0..30]]; // _G. C. Greubel_, Dec 01 2017

%o (Python)

%o def a(n): return (2*n+1)*(5*n**2+5*n+3)//3

%o print([a(n) for n in range(40)]) # _Michael S. Branicky_, Jan 13 2021

%Y (1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

%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.

%Y Cf. A100171, A100174, A051673.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_