OFFSET
0,3
COMMENTS
A strong triangulation is one in which no interior edge joins two vertices of the square (see W. G. Brown reference).
If the restriction "strong" is dropped, the counting sequence is A197271 (shifted left).
LINKS
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14, Issue 4, (1964) 746-768.
William T. Tutte, A census of planar triangulations (Eq. 5.12), Canad. J. Math. 14 (1962), 21-38.
FORMULA
a(n) = 72 * (4*n+3)!/((3*n+6)!*(n-1)!) = 24 * binomial(4*n+3,n-1)/((3*n+5)*(n+2)) = binomial(4*n+3,n-1) - 5 * binomial(4*n+3,n-2) + 6 * binomial(4*n+3,n-3).
EXAMPLE
The 6 triangulations for n=2 are as follows. Four have a central vertex joined to all 4 vertices of the square creating 4 triangular regions, one of which contains the second interior vertex. In these 4 cases, the central vertex has degree 5, the other interior vertex has degree 3. In the other 2 triangulations, both interior vertices have degree 4, an opposite pair a, c of vertices of the square both have degree 3 (so 1 interior edge), and the other 2 opposite vertices have degree 4.
MAPLE
A242136:=n->24*binomial(4*n+3, n-1)/((3*n+5)*(n+2)): seq(A242136(n), n=0..30); # Wesley Ivan Hurt, Aug 16 2014
MATHEMATICA
Table[24 Binomial[4n+3, n-1]/((3n+5)(n+2)), {n, 0, 15}]
CROSSREFS
KEYWORD
nonn
AUTHOR
David Callan, Aug 15 2014
STATUS
approved