OFFSET
0,2
COMMENTS
Number of regions in all the dissections of a convex (n+3)-gon by non-intersecting diagonals. a(1)=5 because in the three dissections of a square we have altogether five regions: one in the "no-diagonals" dissection and two in each of the dissections by one of the two diagonals of the square. - Emeric Deutsch, Dec 28 2003
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions
D. Merlini, R. Sprugnoli and M. C. Verri, Waiting patterns for a printer, FUN with algorithm'01, Isola d'Elba, 2001.
FORMULA
G.f.: (1-z)^2/(8*z^2*sqrt(1-6*z+z^2))-(1+z)/(8*z^2). - Emeric Deutsch, Dec 28 2003
a(n) = T(n+1, n+2), array T as in A049600.
Form an array with the m(n,1)=1 and m(1,n) = n*(n+1)/2 for n=1,2,3... The interior terms m(i,j) = m(i,j-1) + m(i-1,j-1) + m(i-1,j). The terms in the main diagonal = {a(n)}. - J. M. Bergot, Dec 01 2012
D-finite with recurrence: (n+2)*a(n) + (7*n+8)*a(n-1) - (7*n-8)*a(n-2) + (n-2)*a(n-3). - R. J. Mathar, Dec 03 2012
a(n) ~ (3+2*sqrt(2))^(n+3/2) / (2^(9/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 12 2014
a(n) = Jypergeometric2F1([-n, n+3]; [1]; -1), which satisfies the recurrence. - Benedict W. J. Irwin, Oct 14 2016
MATHEMATICA
CoefficientList[Series[(1-x)^2/(8*x^2*Sqrt[1-6*x+x^2])-(1+x)/(8*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
With[{P = LegendreP}, Table[((n+1)*(n+3)*P[n+3, 3] -(6*n^2+22*n+17)*P[n+2, 3] +(n+ 2)*(5*n+8)*P[n+1, 3])/(8*(n+1)*(n+2)), {n, 0, 40}]] (* G. C. Greubel, Oct 20 2022 *)
PROG
(Magma) I:=[1, 5, 26]; [n le 3 select I[n] else ( (7*n+1)*Self(n-1) - (7*n-15)*Self(n-2) + (n-3)*Self(n-3) )/(n+1): n in [1..30]]; // G. C. Greubel, Oct 20 2022
(SageMath)
def A001850(n): return gen_legendre_P(n, 0, 3)
def A035029(n): return ((n+1)*(n+3)*A001850(n+3) - (6*n^2 +22*n +17)*A001850(n+2) + (n+2)*(5*n+8)*A001850(n+1))/(8*(n+1)*(n+2))
[A035029(n) for n in range(40)] # G. C. Greubel, Oct 20 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved