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%I M4115 #201 Apr 27 2024 14:37:16
%S 1,6,18,38,66,102,146,198,258,326,402,486,578,678,786,902,1026,1158,
%T 1298,1446,1602,1766,1938,2118,2306,2502,2706,2918,3138,3366,3602,
%U 3846,4098,4358,4626,4902,5186,5478,5778,6086,6402,6726,7058,7398,7746,8102,8466
%N Number of points on surface of octahedron; also coordination sequence for cubic lattice: a(0) = 1; for n > 0, a(n) = 4n^2 + 2.
%C Also, the number of regions the plane can be cut into by two overlapping concave (2n)-gons. - _Joshua Zucker_, Nov 05 2002
%C 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
%C Binomial transform of a(n) is A055580(n). - _Wesley Ivan Hurt_, Apr 15 2014
%C 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
%C 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
%C Also, coordination sequence for "tfs" 3D uniform tiling. - _N. J. A. Sloane_, Feb 10 2018
%C 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
%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 (225) cF8
%D B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tilings #16 and #22.
%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 T. D. Noe, <a href="/A005899/b005899.txt">Table of n, a(n) for n = 0..1000</a>
%H Barry Balof, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Balof/balof19.html">Restricted tilings and bijections</a>, J. Integer Seq. 15 (2012), no. 2, Article 12.2.3, 17 pp.
%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 Pierre de la Harpe, <a href="https://arxiv.org/abs/2106.02499">On the prehistory of growth of groups</a>, arXiv:2106.02499 [math.GR], 2021.
%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 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%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 Carlos I. Perez-Sanchez, <a href="https://arxiv.org/abs/2401.03705">The Spectral Action on quivers</a>, arXiv:2401.03705 [math.RT], 2024.
%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/pcu">The pcu tiling (or net)</a>
%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/nets/tfs">The tfs tiling (or net)</a>
%H B. K. Teo and N. J. A. Sloane, <a href="http://neilsloane.com/doc/magic1/magic1.html">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985),4545-4558.
%H N. J. A. Sloane, <a href="/A005899/a005899.png">Illustration of a(0)=1, a(1)=6, a(2)=18</a> (from Teo-Sloane 1985)
%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-x))^3. - _Simon Plouffe_ in his 1992 dissertation
%F Binomial transform of [1, 5, 7, 1, -1, 1, -1, 1, ...]. - _Gary W. Adamson_, Nov 02 2007
%F 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
%F Recurrence: n*a(n) = (n-2)*a(n-2) + 6*a(n-1), a(0)=1, a(1)=6. - _Fung Lam_, Apr 15 2014
%F For n > 0, a(n) = A001844(n-1) + A001844(n) = (n-1)^2 + 2n^2 + (n+1)^2. - _Doug Bell_, Aug 18 2015
%F For n > 0, a(n) = A010014(n) - A195322(n). - _R. J. Cano_, Sep 29 2015
%F For n > 0, a(n) = A000384(n+1) + A014105(n-1). - _Bruce J. Nicholson_, Oct 08 2017
%F a(n) = A008574(n) + A008574(n-1) + a(n-1). - _Bruce J. Nicholson_, Dec 18 2017
%F a(n) = 2*d*Hypergeometric2F1(1-d, 1-n, 2, 2) where d=3, n>0. - _Shel Kaphan_, Feb 16 2023
%F a(n) = A035597(n)*3/n, for n>0. - _Shel Kaphan_, Feb 26 2023
%F E.g.f.: exp(x)*(2 + 4*x + 4*x^2) - 1. - _Stefano Spezia_, Mar 08 2023
%F Sum_{n>=0} 1/a(n) = 3/4 + Pi *sqrt(2)*coth( Pi/sqrt 2)/8 = 1.31858... - _R. J. Mathar_, Apr 27 2024
%p A005899:=n->4*n^2 + 2; seq(A005899(n), n=0..50); # _Wesley Ivan Hurt_, Apr 15 2014
%t Join[{1},4Range[40]^2+2] (* or *) Join[{1},LinearRecurrence[{3,-3,1},{6,18,38},40]] (* _Harvey P. Dale_, Nov 08 2011 *)
%o (PARI) Vec(((1+x)/(1-x))^3 + O(x^100)) \\ _Altug Alkan_, Oct 26 2015
%o (Magma) [4*n^2 + 2 : n in [0..50]]; // _Wesley Ivan Hurt_, Oct 26 2015
%Y Partial sums give A001845.
%Y Column 2 * 2 of array A188645.
%Y Cf. A001844, A002522, A008586, A010014, A055580, A195322, A206399.
%Y Cf. A000384, A014105, A000217, A008574, A008412, A035597.
%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 Row 3 of A035607, A266213, A343599.
%Y Column 3 of A113413, A119800, A122542.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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