

A007173


Number of simplicial 3clusters with n cells.
(Formerly M3401)


24



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
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OFFSET

1,4


COMMENTS

Also arises in enumeration of stereoisomers of alkane systems.
"A simplicial dcluster may be informally described as being constructed by gluing regular dsimplexes together facetbyfacet, 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 onetoone correspondence with the combinatorial types of stack polytopes." [Hering et al., 1982]  Jonathan Vos Post, Apr 22 2011
Number of oriented polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}. For oriented polyominoes, chiral pairs are counted as two.  Robert A. Russell, Mar 20 2024


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS



FORMULA

a(n) = C(3n,n)/(3*(2n+1)*(2n+2)) + ([0==n mod 2]*C(3n/2,n) + [1==n mod 2]*C((3n1)/2,(n1)/2)) / (2n+2) + 2*([1==n mod 3]*C(n,(n1)/3) + [2==n mod 3]*C(n,(n2)/3)) / (3n).
a(n) = H(3,n) in Table 8 of Hering link.
G.f.: (8 + 4*G(z)  2*G(z)^2 + z*G(z)^4 + 6*G(z^2) + 3z*G(z^2)^2 + 8z*G(z^3) + 4z^2*G(z^3)^2)/12, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. (End)


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.


KEYWORD

nonn,nice,easy


AUTHOR



EXTENSIONS



STATUS

approved



