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Search: id:A005893
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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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+0
5
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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; listen)
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OFFSET
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0,2
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COMMENT
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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]
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
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
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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).
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
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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]
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MAPLE
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A005893:=-(z+1)*(1+z**2)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
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CROSSREFS
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Cf. A000292, A053545.
Cf. A000217. [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 30 2009]
Sequence in context: A099589 A008141 A119651 this_sequence A008131 A008132 A008115
Adjacent sequences: A005890 A005891 A005892 this_sequence A005894 A005895 A005896
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000
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