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
Also, coordination sequence for "tfs" 3D uniform tiling. - N. J. A. Sloane, Feb 10 2018
Also, the number of n-th order specular reflections arriving at a receiver point from an emitter point inside a cuboid with reflective faces. - Michael Schutte, Sep 18 2018
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
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tilings #16 and #22.
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).
Pierre de la Harpe, On the prehistory of growth of groups, arXiv:2106.02499 [math.GR], 2021.
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
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
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]
Carlos I. Perez-Sanchez, The Spectral Action on quivers, arXiv:2401.03705 [math.RT], 2024.
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 pcu tiling (or net)
Reticular Chemistry Structure Resource (RCSR), The tfs tiling (or net)
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558.
N. J. A. Sloane, Illustration of a(0)=1, a(1)=6, a(2)=18 (from Teo-Sloane 1985)
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(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=1, a(1)=6, a(2)=18, a(3)=38. - 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(n) = 2*d*Hypergeometric2F1(1-d, 1-n, 2, 2) where d=3, n>0. - Shel Kaphan, Feb 16 2023
a(n) = A035597(n)*3/n, for n>0. - Shel Kaphan, Feb 26 2023
E.g.f.: exp(x)*(2 + 4*x + 4*x^2) - 1. - Stefano Spezia, Mar 08 2023
Sum_{n>=0} 1/a(n) = 3/4 + Pi *sqrt(2)*coth( Pi/sqrt 2)/8 = 1.31858... - R. J. Mathar, Apr 27 2024
MAPLE
MATHEMATICA
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.
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.
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved