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A027610
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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.
(Formerly M2688)
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18
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Also arises in enumeration of spectral isomers of alkane systems (see Cyvin et al.). - N. J. A. Sloane (njas(AT)research.att.com), 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.
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REFERENCES
| L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
S. J. Cyvin et al., Enumeration of staggered conformers of alkanes: complete solution ..., J. Molec. Struct., 413 (1997), 237-239.
Hering, F.; Read, R. C.; Shephard, G. C.; The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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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;
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CROSSREFS
| Sum of A047776, A047775, A047774, A047773, A047762, A047760, A047758, A047754, A047753, A047752, A047751, A047771, A047769, A047766 (twice), A047765, A047764.
Sequence in context: A038169 A176606 A007172 * A135688 A005642 A019055
Adjacent sequences: A027607 A027608 A027609 * A027611 A027612 A027613
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Gordon Royle (gordon(AT)maths.uwa.edu.au)
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