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

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

Author

Gordon F. Royle

Keywords

• nonn
• easy
• nice

Comments

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

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.
• 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.

Crossrefs

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}.

Programs

Maple

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;

Mathematica

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 *)

PARI

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

Formula

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)

Extensions

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