A007175 -id:A007175 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

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.

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}.

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 *)

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)

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

A182322 Number of simplicial 4-clusters with n cells. (Formerly M2679).

Original entry on oeis.org

1, 1, 1, 3, 7, 30, 131, 795, 5152, 36800, 272093, 2077909, 16176607, 127996683, 1025727646, 8310377720, 67967600763, 560527576100, 4656993996246, 38949328897318, 327718211568300, 2772480181758683, 23571996461405321, 201327668784954950, 1726755218246463325

Offset: 1

Robert A. Russell, Apr 24 2012

Some of the terms in the Hering article are in error, including the 6th, 8th and 9th.

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

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

This is the average of A007175 and A182299, both of which have Mathematica programs.

a(n) = (A007175(n) + A182299(n))/2.

A182299 Number of achiral simplicial 4-clusters with n cells.

Original entry on oeis.org

1, 1, 1, 3, 6, 20, 51, 184, 550, 2009, 6487, 23875, 81724, 302954, 1078409, 4034373, 14771551, 55789188, 208526682, 794933818, 3017839193, 11604938152, 44590911769, 172833268057, 670520982414, 2617397888002, 10234831661388, 40204487779050, 158254659096516, 625142808049902

Offset: 1

Robert A. Russell, Apr 23 2012

This sequence would be H_{4,n} subtracted from twice h_{4,n} in Table 8 of the Hering article if those numbers were correct, but some are not. In addition, the formula in the penultimate line of Table 5 of the article should not have an exponent for the second E_4.

The limit supremum as n approaches infinity of a(n+1)/a(n) is 16(2-sqrt(3)) or about 4.28719. - Robert A. Russell, Oct 21 2014

			For n=4 the a(4)=3 solutions are the three achiral (there are no chiral) clusters that can be formed from four simplexes in four-space.  One has three attached to a fourth, one has four sharing a common triangle, and the last has neither of these properties.

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

Cf. A182322, A007175.

n = 30;
e[d_,t_]:=Sum[Binomial[d k,k]/((d-1)k+1)t^k,{k,0,n}]
CoefficientList[Series[(10e[4,t^2]e[2,e[4,t^2]t]^3t
   +30e[4,t^4]t(1+e[4,t^4]t)
   +20e[1,e[4,t^6] t^2]e[2,e[4,t^6]t^3]t)/60
   -(6(e[2,e[4,t^2]t]-1)^2+6e[4,t^4]^2t^2)/24
   +(4e[4,t^2]^4t^2+8e[1,e[4,t^6]t^2]e[4,t^6]t^2)/24,
   {t,0,n}]/t,t] (* Robert A. Russell, Apr 23 2012 *)

A369474 Number of chiral pairs of polyominoes composed of n pentachoral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,3,oo}.

Original entry on oeis.org

0, 0, 0, 0, 1, 10, 80, 611, 4602, 34791, 265606, 2054034, 16094883, 127693729, 1024649237, 8306343347, 67952829212, 560471786912, 4656785469564, 38948533963500, 327715193729107, 2772468576820531

Offset: 1

Robert A. Russell, Mar 20 2024

nonn

Also number of chiral pairs of simplicial 4-clusters or stack polytopes with n pentachoral cells. Each member of a chiral pair is a reflection but not a rotation of the other. Some of the h(4,n) terms in the Hering article are in error, including the 6th, 8th and 9th.

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

Cf. A007175 (oriented), A182322 (oriented), A182299 (achiral), A002293 (rooted), A371350 {3,3,oo}.

This is the half the difference of A007175 and A182299, both of which have Mathematica programs.

a(n) = A007175(n) - A182322(n) = (A007175(n) - A182299(n))/2 = A182322(n) - A182299(n).

a(n) = h(4,n) - H(4,n) in Table 8 of Hering link.

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A007173 Number of simplicial 3-clusters with n cells.

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A182322 Number of simplicial 4-clusters with n cells. (Formerly M2679).

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A182299 Number of achiral simplicial 4-clusters with n cells.

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A369474 Number of chiral pairs of polyominoes composed of n pentachoral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,3,oo}.

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