%I M3380 #177 Jun 11 2024 09:33:03

%S 1,4,10,20,34,52,74,100,130,164,202,244,290,340,394,452,514,580,650,

%T 724,802,884,970,1060,1154,1252,1354,1460,1570,1684,1802,1924,2050,

%U 2180,2314,2452,2594,2740,2890,3044,3202,3364,3530,3700,3874,4052,4234

%N Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).

%C Number of n-matchings of the wheel graph W_{2n} (n > 0). Example: a(2)=10 because in the wheel W_4 (rectangle ABCD and spokes OA,OB,OC,OD) we have the 2-matchings: (AB, OC), (AB, OD), (BC, OA), (BC,OD), (CD,OA), (CD,OB), (DA,OB), (DA,OC), (AB,CD) and (BC,DA). - _Emeric Deutsch_, Dec 25 2004

%C For n > 0 a(n) is the difference of two tetrahedral (or pyramidal) numbers: binomial(n+3, 3) = (n+1)(n+2)(n+3)/6. a(n) = A000292(n+1) - A000292(n-3) = (n+1)(n+2)(n+3)/6 - (n-3)(n-2)(n-1)/6. - _Alexander Adamchuk_, May 20 2006; updated by _Peter Munn_, Aug 25 2017 due to changed offset in A000292

%C Equals binomial transform of [1, 3, 3, 1, -1, 1, -1, 1, -1, 1, ...]. Binomial transform of A005893 = nonzero terms of A053545: (1, 5, 19, 63, 191, ...). - _Gary W. Adamson_, Apr 28 2008

%C Disregarding the terms < 10, the sums of four consecutive triangular numbers (A000217). - _Rick L. Shepherd_, Sep 30 2009

%C Use a set of n concentric circles where n >= 0 to divide the plane. a(n) is the maximal number of regions after the 2nd division. - _Frank M Jackson_, Sep 07 2011

%C Euler transform of length 4 sequence [4, 0, 0, -1]. - _Michael Somos_, May 14 2014

%C Also, growth series for affine Coxeter group (or affine Weyl group) A_3 or D_3. - _N. J. A. Sloane_, Jan 11 2016

%C For n > 2 the generalized Pell's equation x^2 - 2*(a(n) - 2)y^2 = (a(n) - 4)^2 has a finite number of positive integer solutions. - _Muniru A Asiru_, Apr 19 2016

%C Union of A188896, A277449, {1,4}. - _Muniru A Asiru_, Nov 25 2016

%C Interleaving of A008527 and A108099. - _Bruce J. Nicholson_, Oct 14 2019

%D N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10b, page 231, W_a(t).

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

%D B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #28.

%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="/A005893/b005893.txt">Table of n, a(n) for n = 0..10000</a>

%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 Steven Edwards and William Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Griffiths/griffiths51.html">On Generalized Delannoy Numbers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.

%H J. M. Grau, C. Miguel, and A. M. Oller-Marcén, <a href="https://arxiv.org/abs/1706.04760">Generalized Quaternion Rings over Z/nZ for an odd n</a>, arXiv:1706.04760 [math.RA], 2017. See Theorem 1, p. 10.

%H Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>. [<a href="/A196265/a196265.pdf">Cached copy</a>.]

%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences </a>, J. Int. Seq. 13 (2010), Article 10.7.8.

%H M. O'Keeffe, <a href="http://dx.doi.org/10.1107/S0108767391006633">N-dimensional diamond, sodalite and rare sphere packings</a>, Acta Cryst., A 47 (1991), 749-753.

%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, <a href="http://rcsr.net/nets/sod">sod</a>.

%H Aditya Sivakumar and Dmitri Tymoczko, <a href="http://dmitri.mycpanel.princeton.edu/homotopy.pdf">Intuitive Musical Homotopy</a>, 2018.

%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. DOI: <a href="http://dx.doi.org/10.1021/ic00220a025">10.1021/ic00220a025</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^4)/(1-x)^4.

%F a(n) = A071619(n-1) + A071619(n) + A071619(n+1), n > 0. - _Ralf Stephan_, Apr 26 2003

%F a(n) = binomial(n+3, 3) - binomial(n-1, 3) for n >= 1. - _Mitch Harris_, Jan 08 2008

%F a(n) = (n+1)^2 + (n-1)^2. - Benjamin Abramowitz, Apr 14 2009

%F a(n) = A000217(n-2) + A000217(n-1) + A000217(n) + A000217(n+1) for n >= 2. - _Rick L. Shepherd_, Sep 30 2009

%F a(n) = 2*n^2 - 0^n + 2. - _Vincenzo Librandi_, Sep 27 2011

%F a(0)=1, a(1)=4, a(2)=10, a(3)=20, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Feb 26 2012

%F a(n) = A228643(n+1,2) for n > 0. - _Reinhard Zumkeller_, Aug 29 2013

%F a(n) = a(-n) for all n in Z. - _Michael Somos_, May 14 2014

%F For n >= 2: a(n) = a(n-1) + 4*n - 2. - _Bob Selcoe_, Mar 22 2016

%F E.g.f.: -1 + 2*(1 + x + x^2)*exp(x). - _Ilya Gutkovskiy_, Apr 19 2016

%F a(n) = 2*A002522(n), n>0. - _R. J. Mathar_, May 30 2022

%F From _Amiram Eldar_, Sep 16 2022: (Start)

%F Sum_{n>=0} 1/a(n) = (coth(Pi)*Pi + 3)/4.

%F Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi)*Pi + 3)/4. (End)

%F Empirical: Integral_{u=-oo..+oo} sigmoid(u)*log(sigmoid(n * u)) du = -Pi^2*a(n) / (24*n), where sigmoid(x) = 1/(1+exp(-x)). Also works for non-integer n>0. - _Carlo Wood_, Dec 04 2023

%F Let P(k,n) be the n-th k-gonal number. Then P(a(k),n) = (k*n-k+1)^2 + (k-1)^2*(n-1). - _Charlie Marion_, May 15 2024

%e G.f. = 1 + 4*x + 10*x^2 + 20*x^3 + 34*x^4 + 52*x^5 + 74*x^6 + 100*x^7 + ...

%p A005893:=-(z+1)*(1+z^2)/(z-1)^3; # _Simon Plouffe_ in his 1992 dissertation

%t Join[{1}, Table[2*(n + 1)^2 + 2, {n, 0, 200}]] (* _Vladimir Joseph Stephan Orlovsky_, Jul 10 2011 *)

%t Join[{1},LinearRecurrence[{3,-3,1},{4,10,20},50]] (* _Harvey P. Dale_, Feb 26 2012 *)

%t a[ n_] := SeriesCoefficient[ (1 - x^4) / (1 - x)^4, {x, 0, Abs@n}]; (* _Michael Somos_, May 14 2014 *)

%t a[ n_] := 2 n^2 + 2 - Boole[n == 0]; (* _Michael Somos_, May 14 2014 *)

%o (Magma) [2*n^2-0^n+2: n in [0..60]]; // _Vincenzo Librandi_, Sep 27 2011

%o (PARI) a(n)=2*n^2-0^n+2 \\ _Charles R Greathouse IV_, Sep 24 2015

%Y Cf. A000217, A000292, A053545, A206399.

%Y Cf. similar sequences listed in A255843.

%Y The growth series for the affine Coxeter groups D_3 through D_12 are A005893 and A266759-A266767.

%Y For partial sums see A005894.

%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_