A047771 -id:A047771 - 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

A027610 The number of Apollonian networks (planar 3-trees) with n+3 vertices.

Original entry on oeis.org

1, 1, 1, 3, 7, 24, 93, 434, 2110, 11002, 58713, 321776, 1792133, 10131027, 57949430, 334970205, 1953890318, 11489753730, 68054102361, 405715557048, 2433003221232, 14668536954744, 88869466378593, 540834155878536, 3304961537938269, 20273202069859769

Offset: 1

Gordon F. Royle

Previous name was: Number of chordal planar triangulations; also number of planar triangulations with maximal number of triangles; also number of graphs obtained from the tetrahedron by repeatedly inserting vertices of degree 3 into a triangular face; also number of uniquely 4-colorable planar graphs; also number of simplicial 3-clusters with n cells; also Apollonian networks with n+3 vertices.
Also arises in enumeration of spectral isomers of alkane systems (see Cyvin et al.). - N. J. A. Sloane, Aug 15 2006
Chordal planar triangulations: take planar triangulations on n nodes, divide them into classes according to how many triangles they contain (all have 2n-4 triangular faces but may have additional triangles); count triangulations in class with most triangles.
If mirror images are not taken as equivalent, A007173 is obtained instead. - Brendan McKay, Mar 08 2014
Number of unoriented polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}. For unoriented polyominoes, chiral pairs are counted as one. - 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.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
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.
Paul Jungeblut, Edge Guarding Plane Graphs, Master Thesis, Karlsruhe Institute of Technology (Germany, 2019).
F. Hering et al., The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.
Manfred Scheucher, Hendrik Schrezenmaier, Raphael Steiner, A Note On Universal Point Sets for Planar Graphs, arXiv:1811.06482 [math.CO], 2018.

Sum of A047776, A047775, A047774, A047773, A047762, A047760, A047758, A047754, A047753, A047752, A047751, A047771, A047769, A047766 (twice), A047765, A047764.

Cf. A007173 (oriented), A371350 (chiral), A371351 (achiral), A001764 (rooted), A000207 {3,oo}, A182322 {3,3,3,oo}.

A001764 := proc(m) RETURN((3*m)!/(m!*(2*m+1)!)); end; # Gives A001764(m)
A047749 := proc(m) local x; if m mod 2 = 1 then x := (m-1)/2; RETURN((3*x+1)!/((x+1)!*(2*x+1)!)); fi; RETURN(A001764(m/2)); end; # Gives A047749(m)
A027610 := proc(n) local N; N := 0; N := N + A001764(n)/(12*(n+1)); if n mod 2 = 0 then N := N + 5/24*A001764(n/2); fi; if (n-1) mod 3 = 0 then N := N + 1/3*A001764((n-1)/3); fi; if (n-1) mod 4 = 0 then N := N + 1/4*A001764((n-1)/4); fi;
if (n-2) mod 6 = 0 then N := N + 1/6*A001764((n-2)/6); fi; N := N + 3/8*A047749(n); if (2*n-1) mod 3 = 0 then N := N + 1/6*A047749((2*n-1)/3); fi; RETURN(N); end;

Table[Binomial[3 n, 2 n]/(6 (2 n + 1) (2 n + 2)) + If[EvenQ[n], 7 Binomial[3 n/2, n]/(12 (n + 1)), 3 Binomial[3 n/2 - 1/2, n]/(4 (n + 1))] + Switch[Mod[n, 3], 1, Binomial[n - 1, 2 n/3 - 2/3]/(2 n/3 + 1/3), 2, Binomial[n - 1, 2 n/3 - 1/3]/(2 n/3 + 2/3), , 0]/3 + If[1 == Mod[n,4], Binomial[3 n/4 - 3/4, n/2 - 1/2]/(n/2 + 1/2), 0]/4 + If[2 == Mod[n, 6], Binomial[n/2 - 1, n/3 - 2/3]/(n/3 + 1/3), 0]/6, {n, 1, 30}] (* _Robert A. Russell, Apr 11 2012 *)

T(m)={if(m<0||denominator(m)!=1,0,(3*m)!/(m!*(2*m+1)!))};
U(k)={if(k<0||denominator(k)!=1,0,if(k%2,my(m=(k-1)/2);(3*m+1)!/((m+1)!*(2*m+1)!),T(k/2)))};
S(n)=T(n)/(12*(n+1))+5*T(n/2)/24+T((n-1)/3)/3+T((n-1)/4)/4+T((n-2)/6)/6+3*U(n)/8+U((2*n-1)/3)/6;
for(k=1,26,print1(S(k),", ")) \\ Hugo Pfoertner, Mar 07 2020

From Robert A. Russell, Mar 20 2024: (Start)
a(n) = C(3n,n)/(6*(2n+1)*(2n+2)) + ([0==n mod 2]*7*C(3n/2,n) + [1==n mod 2]*9*C((3n-1)/2,n)) / (12(n+1)) + [1==n mod 3]*C(n-1,(n-1)/3)/(2n+1) + [2==n mod 3]*C(n-1,(n-2)/3)/(2n+2) + [1==n mod 4]*C((3n-3)/4,(n-1)/2)/(2n+2) + [2==n mod 6]*C(n/2-1,(n-2)/3)/(2n+2).
a(n) = A007173(n) - A371350(n) = (A007173(n) + A371351(n))/2 = A371350(n) + A371351(n).
a(n) = h(3,n) in Table 8 of Hering link.
G.f.: (-16 + 4*G(z) - 2*G(z)^2 + z*G(z)^4 + 14*G(z^2) + 9z*G(z^2)^2 + 8z*G(z^3) + 4z^2*G(z^3)^2 + 6z*G(z^4) + 4z^2*G(z^6))/24, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. (End)

One additional term from Robert A. Russell, Apr 11 2012
Noted the name "Apollonian network" by Brendan McKay, Mar 08 2014
New name from Allan Bickle, Feb 21 2024

A371351 Number of achiral polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 15, 37, 73, 182, 364, 952, 1944, 5169, 10659, 28842, 60115, 164450, 345345, 953814, 2016144, 5609760, 11920740, 33378072, 71250060, 200553733, 429757960, 1215177680, 2612635888, 7416503776

Offset: 1

Robert A. Russell, Mar 19 2024

nonn

Also number of achiral simplicial 3-clusters or stack polytopes with n tetrahedral cells. An achiral polyomino is identical to its reflection.

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

Cf. A007173 (oriented), A027610 (oriented), A371350 (chiral), A001764 (rooted), A208355(n-1) {3,oo}, A182299 {3,3,3,oo}.

Table[(If[OddQ[n],3Binomial[(3n-1)/2,n],2Binomial[3n/2,n]]+If[1==Mod[n,4],3Binomial[(3n-3)/4,(n-1)/2],0]+If[2==Mod[n,6],3Binomial[n/2-1,(n-2)/3],0])/(3n+3),{n,30}]

a(n) = ([0==n mod 2]*2*C(3n/2,n) + [1==n mod 2]*3*C((3n-1)/2,n) + [1==n mod4]*3*C((3n-3)/4,(n-1)/2) + [2==n mod6]*3*C(n/2-1,(n-2)/3)) / (3n+3).
a(n) = 2*A027610(n) - A007173(n) = A007173(n) - 2*A371350(n) = A027610(n) - A371350(n).
a(n) = 2*H(3,n) - h(3,n) in Table 8 of Hering link.
G.f.: (-4 + 4*G(z^2) + 3z*G(z^2)^2 + 3z*G(z^4) + 2z^2*G(z^6)) / 6, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764.

A047776 Number of chiral pairs of asymmetric dissectable polyhedra with n tetrahedral cells (type A).

Original entry on oeis.org

0, 0, 0, 0, 2, 11, 71, 370, 2005, 10682, 58167, 320116, 1789210, 10121965, 57933469, 334919626, 1953800059, 11489466014, 68053583772, 405713887061, 2433000197471, 14668527134167, 88869448492895, 540834097467624, 3304961431043989, 20273201718862728, 124798671079300720, 770762029389852807

Offset: 1

N. J. A. Sloane

One of 17 different symmetry types comprising A007173 and A027610 and one of 7 for A371350. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both asymmetric (type A) with n tetrahedral cells. The order of the symmetry group is 1. Each member of a chiral pair is a reflection but not a rotation of the other. - Robert A. Russell, Mar 31 2024

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
Robert A. Russell, Mathematica Graphics3D program for A047776 examples

Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047775 (type B), A047774 (type C). A047773 (type D), A047762 (type E), A047760 (type F), A047758 (type G), A047754 (type H), A047753 (type I), A047752 (type J), A047751 (type K), A047771 (type L), A047769 (type M), A047766 (type N|O), A047765 (type P), A047764 (type Q).

Table[If[n < 5, 0, Binomial[3 n, 2 n + 2]/(3 n (n - 1))
    - If[OddQ[n], Binomial[3 n/2 - 1/2, n + 1] 3/(n - 1),
     7 Binomial[3 n/2, n + 1]/(3 n)]
    - Switch[Mod[n, 3], 1, Binomial[n - 1, 2 n/3 + 1/3]/(n - 1), 2,
     Binomial[n - 1, 2 n/3 + 2/3]/(n - 2), _, 0]
    + Switch[Mod[n, 4], 1, Binomial[3 n/4 - 3/4, n/2 + 1/2] 2/(3 (n - 1))
      + Binomial[3 n/4 + 1/4, n/2 + 3/2] 4/(n - 1) +
      Binomial[3 n/4 - 3/4, n/2 + 1/2] 4/(n + 3), 2,
     Binomial[3 n/4 - 1/2, n/2 + 1] 8/(n - 2), 3,
     Binomial[3 n/4 - 1/4, n/2 + 3/2] 12/(n - 3), 0,
     Binomial[3 n/4 - 1, n/2 + 1] 12/(n - 4)] +
    Switch[Mod[n, 6], 1, Binomial[n/2 - 1/2, n/3 + 2/3] 6/(n - 1), 2,
     Binomial[n/2 - 1, n/3 + 1/3] 4/(n - 2) +
      Binomial[n/2, n/3 + 4/3] 6/(n - 2) +
      Binomial[n/2 - 1, n/3 + 1/3] 6/(n + 4), 4,
     Binomial[n/2 - 1, n/3 + 2/3] 12/(n - 4), 5,
     Binomial[n/2 - 1/2, n/3 + 1/3] 9/(n + 4), _, 0] +
    Switch[Mod[n, 12], 2, -Binomial[n/4 - 1/2, n/6 + 2/3] 12/(n - 2), 5,
     Binomial[n/4 - 5/4, n/6 - 5/6] 2/(n + 1),
     8, -Binomial[n/4 - 1, n/6 - 1/3] 12/(n + 4), _, 0] -
    Switch[Mod[n, 24], 5, Binomial[n/8 - 5/8, n/12 - 5/12] 12/(n + 7), 17,
     Binomial[n/8 - 9/8, n/12 - 5/12] 24/(n + 7), , 0]]/2, {n, 1, 60}] (* _Robert A. Russell, Apr 09 2012 *)

From Robert A. Russell, Mar 31 2024: (Start)
a(n) = A001764(n)/(12(n+1)) - A047775(n)/2 - A047774(n)/3 - A047773(n)/6 - A047762(n)/2 - A047760(n)/4 - A047758(n)/4 - A047754(n)/4 - A047753(n)/8 - A047752(n)/12 - A047751(n)/24 - A047771(n)/2 - A047769(n)/2 - A047766(n)/6 - A047766(n)/6 - A047765(n)/4 - A047764(n)/12.
G.f.: (G(z^4) + G(z^6) - 2)/(2z) - z/3 + G(z)/6 - G(z)^2/12 + z*G(z)^4/24 - 7*G(z^2)/12 - 3z*G(z^2)^2/8 - z*G(z^3)/6 - z^2*G(z^3)^2/12 + G(z^4)/2 - z*G(z^4)/6 + (z*G(z^4)^2 + z^2*G(z^4)^2 + z*G(z^6))/2 + z^2*G(z^6)/12 + (z^2*G(z^6)^2 + z^4*G(z^6)^2 - z^2*G(z^12))/2 + z^5*G(z^12)/6 - (z^8*G(z^12)^2 + z^5*G(z^24) + z^17*G(z^24)^2)/2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. (End)

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A047772 Dissections: bisection of A047771.

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A007173 Number of simplicial 3-clusters with n cells.

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A027610 The number of Apollonian networks (planar 3-trees) with n+3 vertices.

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A371351 Number of achiral polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}.

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A047776 Number of chiral pairs of asymmetric dissectable polyhedra with n tetrahedral cells (type A).

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