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A007173 Number of simplicial 3-clusters with n cells.
(Formerly M3401)
3
1, 1, 1, 4, 10, 40, 171, 831, 4147, 21822, 117062, 642600, 3582322, 20256885, 115888201, 669911568, 3907720521, 22979343010, 136107859377, 811430160282, 4866004426320, 29337068299728, 177738920836446, 1081668278379000, 6609923004626478, 40546403939165805 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Also arises in enumeration of stereoisomers of alkane systems.

"A simplicial d-cluster may be informally described as being constructed by gluing regular d-simplexes together facet-by-facet, at each stage gluing a new simplex to exactly one facet of a cluster already constructed. The equivalence classes of such clusters under rigid motions are in one-to-one correspondence with the combinatorial types of stack polytopes." [Hering et al., 1982] - Jonathan Vos Post, Apr 22 2011

Hering article has error in the 14th term. - Robert A. Russell, Apr 11 2012

Also same as A027610 with mirror-image not treated as equivalence. - Brendan McKay, Mar 08 2014

REFERENCES

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 = 1..200

L. W. Beineke and R. E. Pippert Enumerating dissectable polyhedra by their automorphism groups, Can. J. Math., 26 (1974), 50-67

S. J. Cyvin et al., Staggered conformers of alkanes: complete solution of the enumeration problem, J. Molec. Struct., 413 (1997), 227-239.

F. Hering et al., The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.

MATHEMATICA

Table[Binomial[3 n, n]/(3 (2 n + 1) (2 n + 2)) + If[OddQ[n], Binomial[3 (n - 1)/2 + 1, n]/(n + 1), Binomial[3 n/2, n]/(n + 1)]/2 + 2 Switch[Mod[n, 3], 0, 0, 1, Binomial[n, (n - 1)/3]/n, 2, Binomial[n, (n - 2)/3]/n]/3, {n, 1, 30}] (* Robert A. Russell, Apr 11 2012 *)

CROSSREFS

Sum of achiral symmetry types (A047775, A047773, A047760, A047754, A047753, A047751, A047771, A047766 [type N], A047765, A047764) plus twice sum of chiral symmetry types (A047776, A047774, A047762, A047758, A047752, A047769, A047766 [type O]) in Beineke article.

Sequence in context: A149212 A223160 A263551 * A222367 A220817 A114918

Adjacent sequences:  A007170 A007171 A007172 * A007174 A007175 A007176

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Corrected a(14) and added additional terms. - Robert A. Russell, Apr 11 2012

STATUS

approved

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Last modified February 19 20:54 EST 2019. Contains 320328 sequences. (Running on oeis4.)