

A087809


Number of triangulations (by Euclidean triangles) having 3+3n vertices of a triangle with each side subdivided by n additional points.


0



1, 4, 29, 229, 1847, 14974, 121430, 983476, 7952111, 64193728, 517447289, 4165721377, 33500374796, 269166095800, 2161064409680, 17339917293304, 139060729285871, 1114752741216196, 8933074352513183, 71564554425680839
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OFFSET

0,2


LINKS

Table of n, a(n) for n=0..19.
Andrei Asinowski, Christian Krattenthaler, Toufik Mansour, Counting triangulations of some classes of subdivided convex polygons, European Journal of Combinatorics 62 (2017), 92114; arXiv:1604.02870 [math.CO], 2016. See also preprint, 2016.
R. Bacher, Counting Triangulations of Configurations, arXiv:math/0310206 [math.CO], 2003.


FORMULA

A formula is given in the Bacher reference.
It seems that a(n) = Sum_{i, j, k>=0} C(n, i+j)*C(n, j+k)*C(n, k+i)).  Benoit Cloitre, Oct 25 2004; proved in the article by Asinowski et al.
G.f.: seems to be (10*g^3  17*g^2 + 7*g  1)/((13*g)*(2*g1)*(4*g^2  6*g+1)) where g*(1g)^2 = x.  Mark van Hoeij, Nov 10 2011; proved in the article by Asinowski et al.
Conjecture: 2*n*(2*n1)*(5*n^2  29*n + 30)*a(n) + (295*n^4 + 1926*n^3  3425*n^2 + 2106*n  360)*a(n1) + 24*(3*n4)*(3*n5)*(5*n^2  19*n + 6)*a(n2) = 0.  R. J. Mathar, Apr 23 2015. Proved by Andrei Asinowski, C. Krattenthaler, T. Mansour, Counting triangulations of balanced subdivisions of convex polygons, 2016.


EXAMPLE

a(0)=1 since there is only one triangulation of a triangle (consisting of the triangle itself).
The a(1)=4 triangulations of a triangle with each side subdivided by one additional point are given by
.
O O
/ \ /\
O _ O O O
/ \ / \ / \/ \
O _ O _ O , O _ O _ O
.
and rotations by 120 degrees and 240 degrees of the last triangulation.


MATHEMATICA

max = 19; f[x_] := Sum[ a[n]*x^n, {n, 0, max}]; a[0] = 1; g[x_] := Sum[ b[n]*x^n, {n, 0, max}]; b[0] = 0; coes = CoefficientList[ Series[ g[x]*(1  g[x])^2  x, {x, 0, max}], x]; solb = Solve[ Thread[ coes == 0]][[1]]; coes = CoefficientList[ Series[ f[x]  ((10*g[x]^3  17*g[x]^2 + 7*g[x]  1)/((1  3*g[x])*(2*g[x]  1)*(4*g[x]^2  6*g[x] + 1))), {x, 0, max}], x] /. solb; sola = Solve[ Thread[ coes == 0]][[1]]; Table[a[n] /. sola, {n, 0, max}] (* JeanFrançois Alcover, Dec 06 2011, after Mark van Hoeij *)


CROSSREFS

Sequence in context: A079756 A344098 A221415 * A140526 A151343 A208812
Adjacent sequences: A087806 A087807 A087808 * A087810 A087811 A087812


KEYWORD

nonn,nice


AUTHOR

Roland Bacher, Oct 16 2003


STATUS

approved



