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 A027610 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. (Formerly M2688) 21

%I M2688

%S 1,1,1,3,7,24,93,434,2110,11002,58713,321776,1792133,10131027,

%T 57949430,334970205,1953890318,11489753730,68054102361,405715557048,

%U 2433003221232,14668536954744,88869466378593,540834155878536,3304961537938269,20273202069859769

%N 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.

%C Also arises in enumeration of spectral isomers of alkane systems (see Cyvin et al.). - _N. J. A. Sloane_, Aug 15 2006

%C 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.

%C If mirror images are not taken as equivalent, A007173 is obtained instead. - _Brendan McKay_, Mar 08 2014

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

%H Vincenzo Librandi, <a href="/A027610/b027610.txt">Table of n, a(n) for n = 1..200</a>

%H L. W. Beineke and R. E. Pippert <a href="http://dx.doi.org/10.4153/CJM-1974-006-x">Enumerating dissectable polyhedra by their automorphism groups</a>, Can. J. Math., 26 (1974), 50-67

%H CombOS - Combinatorial Object Server, <a href="http://combos.org/plantri">generate planar graphs</a>

%H S. J. Cyvin, Jianji Wang, J. Brunvoll, Shiming Cao, Ying Li, B. N. Cyvin, and Yugang Wang, <a href="https://doi.org/10.1016/S0022-2860(97)00025-2">Staggered conformers of alkanes: complete solution of the enumeration problem</a>, J. Molec. Struct. 413-414 (1997), 227-239.

%H Paul Jungeblut, <a href="https://i11www.iti.kit.edu/_media/teaching/theses/ma-jungeblut-19.pdf">Edge Guarding Plane Graphs</a>, Master Thesis, Karlsruhe Institute of Technology (Germany, 2019).

%H F. Hering et al., <a href="http://dx.doi.org/10.1016/0012-365X(82)90121-2">The enumeration of stack polytopes and simplicial clusters</a>, Discrete Math., 40 (1982), 203-217.

%H Manfred Scheucher, Hendrik Schrezenmaier, Raphael Steiner, <a href="https://arxiv.org/abs/1811.06482">A Note On Universal Point Sets for Planar Graphs</a>, arXiv:1811.06482 [math.CO], 2018.

%p A001764 := proc(m) RETURN((3*m)!/(m!*(2*m+1)!)); end; # Gives A001764(m)

%p 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)

%p 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;

%p 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;

%t 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 *)

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

%K nonn,easy,nice,changed

%O 1,4

%A _Gordon F. Royle_

%E One additional term from _Robert A. Russell_, Apr 11 2012

%E Noted the name "Apollonian network" by _Brendan McKay_, Mar 08 2014

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Last modified February 22 12:33 EST 2020. Contains 332136 sequences. (Running on oeis4.)