%I A005893 M3380
%S A005893 1,4,10,20,34,52,74,100,130,164,202,244,290,340,394,452,514,580,650,
%T A005893 724,802,884,970,1060,1154,1252,1354,1460,1570,1684,1802,1924,2050,
%U A005893 2180,2314,2452,2594,2740,2890,3044,3202,3364,3530,3700,3874,4052,4234
%N A005893 Number of points on surface of tetrahedron: 2n^2 + 2 (coordination sequence 
               for sodalite net) for n>0.
%C A005893 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 
               (deutsch(AT)duke.poly.edu), Dec 25 2004
%C A005893 For n>0 a(n) is the difference of two tetrahedral(or pyramidal) numbers: 
               C(n+3,3) = (n+1)(n+2)(n+3)/6. a(n) = A000292(n) + A000292(n-4) = 
               (n+1)(n+2)(n+3)/6 - (n-3)(n-2)(n-1)/6. - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               May 20 2006
%C A005893 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 (qntmpkt(AT)yahoo.com), Apr 28 2008
%C A005893 Disregarding the terms < 10, the sums of four consecutive triangular 
               numbers (A000217). [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), 
               Sep 30 2009]
%D A005893 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005893 H. S. M. Coxeter, ``Polyhedral numbers,'' in R. S. Cohen et al., editors, 
               For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
%D A005893 R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. 
               Anchor, NY, 1973, p. 46.
%D A005893 M. O'Keeffe, N-dimensional diamond, sodalite and rare sphere packings, 
               Acta Cryst., A 47 (1991), 749-753.
%D A005893 B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral 
               clusters, Inorgan. Chem. 24 (1985), 4545-4558.
%H A005893 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:/
               /www.research.att.com/~njas/doc/ldl7.txt">Abstract</a>, <a href="http:/
               /www.research.att.com/~njas/doc/ldl7.pdf">pdf</a>, <a href="http:/
               /www.research.att.com/~njas/doc/ldl7.ps">ps</a>).
%H A005893 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005893 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A005893 Expansion of (1-x^4 )/(1-x)^4.
%F A005893 a(n) = A071619(n-1) + A071619(n) + A071619(n+1), n>0. - Ralf Stephan, 
               Apr 26 2003
%F A005893 a(n) = C(n+3,3) - C(n-1,3) for n >= 1. - Mitch Harris (maharri(AT)gmail.com), 
               Jan 08 2008
%F A005893 a(n) = (n+1)^2 + (n-1)^2. - Benjamin Abramowitz, Apr 14 2009
%F A005893 a(n) = A000217(n-2) + A000217(n-1) + A000217(n) + A000217(n+1) for n 
               >= 2. [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 30 
               2009]
%p A005893 A005893:=-(z+1)*(1+z**2)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
%Y A005893 Cf. A000292, A053545.
%Y A005893 Cf. A000217. [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 
               30 2009]
%Y A005893 Sequence in context: A099589 A008141 A119651 this_sequence A008131 A008132 
               A008115
%Y A005893 Adjacent sequences: A005890 A005891 A005892 this_sequence A005894 A005895 
               A005896
%K A005893 nonn,easy,nice
%O A005893 0,2
%A A005893 N. J. A. Sloane (njas(AT)research.att.com).
%E A005893 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000