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A087809
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Number of triangulations (by Euclidean triangles) having 3+3n vertices of a triangle with each side subdivided by n additional points.
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0
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1, 4, 29, 229, 1847, 14974, 121430, 983476, 7952111, 64193728, 517447289, 4165721377, 33500374796, 269166095800, 2161064409680, 17339917293304, 139060729285871, 1114752741216196, 8933074352513183, 71564554425680839
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| R. Bacher, Counting Triangulations of Configurations, arXiv:math.CO/0310206, http://fr.arXiv.org/abs/math.CO/0310206
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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
G.f.: seems to be (10*g^3-17*g^2+7*g-1)/((1-3*g)*(2*g-1)*(4*g^2-6*g+1)) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011
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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.
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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}] (* From Jean-François Alcover, Dec 06 2011, after Mark van Hoeij *)
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CROSSREFS
| Sequence in context: A001883 A135429 A079756 * A140526 A151343 A125808
Adjacent sequences: A087806 A087807 A087808 * A087810 A087811 A087812
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KEYWORD
| nonn,nice
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AUTHOR
| Roland Bacher (roland.bacher(AT)ujf-grenoble.fr), Oct 16 2003
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