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A005893
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Number of points on surface of tetrahedron: 2n^2 + 2 (coordination sequence for sodalite net) for n>0.
(Formerly M3380)
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12
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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 (deutsch(AT)duke.poly.edu), Dec 25 2004
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
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
Disregarding the terms < 10, the sums of four consecutive triangular numbers (A000217). [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), 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
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REFERENCES
| H. S. M. Coxeter, ``Polyhedral numbers,'' in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.
M. O'Keeffe, N-dimensional diamond, sodalite and rare sphere packings, Acta Cryst., A 47 (1991), 749-753.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
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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 (Abstract, pdf, ps).
Guo-Niu Han, Enumeration of Standard Puzzles
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| Expansion of (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) = C(n+3,3) - C(n-1,3) for n >= 1. - Mitch Harris (maharri(AT)gmail.com), 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. [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 30 2009]
a(n) = 2*n^2-0^n+2. - Vincenzo Librandi, Sep 27 2011
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MAPLE
| A005893:=-(z+1)*(1+z**2)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Join[{1}, Table[2*(n + 1)^2 + 2, {n, 0, 200}]] (* From Vladimir Joseph Stephan Orlovsky, Jul 10 2011 *)
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PROG
| (MAGMA) [2*n^2-0^n+2: n in [0..60]]; // Vincenzo Librandi, Sep 27 2011
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CROSSREFS
| Cf. A000217, A000292, A053545, A206399.
Sequence in context: A099589 A008141 A119651 * A008131 A008132 A008115
Adjacent sequences: A005890 A005891 A005892 * A005894 A005895 A005896
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000
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