A007173 Number of simplicial 3-clusters with n cells.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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

The Hering article has an 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

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

• 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
• CombOS - Combinatorial Object Server, generate planar graphs
• S. J. Cyvin, Jianji Wang, J. Brunvoll, Shiming Cao, Ying Li, B. N. Cyvin, and Yugang Wang, Staggered conformers of alkanes: complete solution of the enumeration problem, J. Molec. Struct. 413-414 (1997), 227-239.
• F. Hering et al., The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.

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.

Cf. A027610 (unoriented), A371350 (chiral), A371351 (achiral), A001764 (rooted), A001683(n+2) {3,oo}, A007175 {3,3,3,oo}.

Programs

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 *)

Formula

From Robert A. Russell, Mar 20 2024: (Start)

a(n) = C(3n,n)/(3*(2n+1)*(2n+2)) + ([0==n mod 2]*C(3n/2,n) + [1==n mod 2]*C((3n-1)/2,(n-1)/2)) / (2n+2) + 2*([1==n mod 3]*C(n,(n-1)/3) + [2==n mod 3]*C(n,(n-2)/3)) / (3n).

a(n) = A027610(n) + A371350(n) = 2*A027610(n) - A371351(n) = 2*A371350(n) + A371351(n).

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)

Extensions

a(14) corrected and additional terms from Robert A. Russell, Apr 11 2012