A005893 Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).

1, 4, 10, 20, 34, 52, 74, 100, 130, 164, 202, 244, 290, 340, 394, 452, 514, 580, 650, 724, 802, 884, 970, 1060, 1154, 1252, 1354, 1460, 1570, 1684, 1802, 1924, 2050, 2180, 2314, 2452, 2594, 2740, 2890, 3044, 3202, 3364, 3530, 3700, 3874, 4052, 4234

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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

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

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

Disregarding the terms < 10, the sums of four consecutive triangular numbers (A000217). - Rick L. Shepherd, Sep 30 2009

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

Euler transform of length 4 sequence [4, 0, 0, -1]. - Michael Somos, May 14 2014

Also, growth series for affine Coxeter group (or affine Weyl group) A_3 or D_3. - N. J. A. Sloane, Jan 11 2016

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

Union of A188896, A277449, {1,4}. - Muniru A Asiru, Nov 25 2016

Interleaving of A008527 and A108099. - Bruce J. Nicholson, Oct 14 2019

Examples

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

References

• 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).
• H. S. M. Coxeter, "Polyhedral numbers," in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
• B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #28.
• 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

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
• Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
• J. M. Grau, C. Miguel, and A. M. Oller-Marcén, Generalized Quaternion Rings over Z/nZ for an odd n, arXiv:1706.04760 [math.RA], 2017. See Theorem 1, p. 10.
• Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy.]
• Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010), Article 10.7.8.
• M. O'Keeffe, N-dimensional diamond, sodalite and rare sphere packings, Acta Cryst., A 47 (1991), 749-753.
• 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, sod.
• Aditya Sivakumar and Dmitri Tymoczko, Intuitive Musical Homotopy, 2018.
• B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558. DOI: 10.1021/ic00220a025.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A000292, A053545, A206399.

Cf. similar sequences listed in A255843.

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

For partial sums see A005894.

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.

Programs

Magma

[2*n^2-0^n+2: n in [0..60]]; // Vincenzo Librandi, Sep 27 2011

Maple

A005893:=-(z+1)*(1+z^2)/(z-1)^3; # Simon Plouffe in his 1992 dissertation

Mathematica

Join[{1}, Table[2*(n + 1)^2 + 2, {n, 0, 200}]] (* Vladimir Joseph Stephan Orlovsky, Jul 10 2011 *)
Join[{1},LinearRecurrence[{3,-3,1},{4,10,20},50]] (* Harvey P. Dale, Feb 26 2012 *)
a[ n_] := SeriesCoefficient[ (1 - x^4) / (1 - x)^4, {x, 0, Abs@n}]; (* Michael Somos, May 14 2014 *)
a[ n_] := 2 n^2 + 2 - Boole[n == 0]; (* Michael Somos, May 14 2014 *)

PARI

a(n)=2*n^2-0^n+2 \\ Charles R Greathouse IV, Sep 24 2015

Formula

G.f.: (1 - x^4)/(1-x)^4.

a(n) = A071619(n-1) + A071619(n) + A071619(n+1), n > 0. - Ralf Stephan, Apr 26 2003

a(n) = binomial(n+3, 3) - binomial(n-1, 3) for n >= 1. - Mitch Harris, Jan 08 2008

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

a(n) = A000217(n-2) + A000217(n-1) + A000217(n) + A000217(n+1) for n >= 2. - Rick L. Shepherd, Sep 30 2009

a(n) = 2*n^2 - 0^n + 2. - Vincenzo Librandi, Sep 27 2011

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

a(n) = A228643(n+1,2) for n > 0. - Reinhard Zumkeller, Aug 29 2013

a(n) = a(-n) for all n in Z. - Michael Somos, May 14 2014

For n >= 2: a(n) = a(n-1) + 4*n - 2. - Bob Selcoe, Mar 22 2016

E.g.f.: -1 + 2*(1 + x + x^2)*exp(x). - Ilya Gutkovskiy, Apr 19 2016

a(n) = 2*A002522(n), n>0. - R. J. Mathar, May 30 2022

From Amiram Eldar, Sep 16 2022: (Start)

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

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

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

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