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A005899
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Number of points on surface of octahedron: a(0) = 1; for n>0, a(n) = 4n^2 + 2; coordination sequence for cubic lattice.
(Formerly M4115)
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14
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1, 6, 18, 38, 66, 102, 146, 198, 258, 326, 402, 486, 578, 678, 786, 902, 1026, 1158, 1298, 1446, 1602, 1766, 1938, 2118, 2306, 2502, 2706, 2918, 3138, 3366, 3602, 3846, 4098, 4358, 4626, 4902, 5186
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Also, the number of regions the plane can be cut into by two overlapping concave (2n)-gons. - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Nov 05 2002
If X is an n-set and Y_i (i=1,2,3) mutually disjoint 2-subsets of X then a(n-5) is equal to the number of 5-subests of X intersecting each Y_i (i=1,2,3). - Milan R. Janjic (agnus(AT)blic.net), Aug 26 2007
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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.
Gmelin Handbook of Inorg. and Organomet. Chem., 8th Ed., 1994, TYPIX search code (225) cF8
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).
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
| T. D. Noe, Table of n, a(n) for n=0..1000
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).
R. W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, Algebraic Description of Coordination Sequences and Exact Topological Densities for Zeolites, Acta Cryst., A52 (1996), pp. 879-889.
R. W. Grosse-Kunstleve, Coordination Sequences and Encyclopedia of Integer Sequences
Milan Janjic, Two Enumerative Functions
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
| G.f.: ((1+x)/(1-x))^3.
Binomial transform of [1, 5, 7, 1, -1, 1, -1, 1,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 02 2007
a(0)=1, a(1)=6, a(2)=18, a(3)=38, a(n)=3*a(n-1)-3*a(n-2)+a(n-3) [From Harvey P. Dale, Nov 08 2011]
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MAPLE
| A005899:=-(z+1)**3/(z-1)**3; [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| s=2; lst={s-1}; Do[s+=n+1; AppendTo[lst, s], {n, 3, 6!, 8}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 25 2008]
Join[{1}, 4Range[40]^2+2] (* or *) Join[{1}, LinearRecurrence[{3, -3, 1}, {6, 18, 38}, 40]] (* From Harvey P. Dale, Nov 08 2011 *)
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CROSSREFS
| Partial sums give A001845.
Column 2 * 2 of array A188645.
Cf. A206399.
Sequence in context: A185223 A101853 A132432 * A180118 A129863 A035489
Adjacent sequences: A005896 A005897 A005898 * A005900 A005901 A005902
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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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