A387224 Number of dissections of a convex n-gon by strictly disjoint diagonals so as to create no triangles.

0, 1, 1, 4, 8, 17, 37, 81, 177, 389, 859, 1905, 4241, 9477, 21251, 47806, 107864, 244045, 553575, 1258687, 2868285, 6549757, 14985361, 34347444, 78860152, 181347591, 417653187, 963234195, 2224464087, 5143567237, 11907471643, 27597112946, 64028244032, 148703128913, 345690623119

Offset: 3

Author

Muhammed Sefa Saydam, Aug 22 2025

Keywords

• nonn
• new

Comments

Strictly disjoint diagonals means that the diagonals are non-crossing and may not share endpoints.

Examples

			         n=4                         n=5                            n=6
    (1)       (2)                    (1)              (1) (2)     (1) (2)     (1) (2)
                                  (5)   (2)         (6)  \  (3) (6)-----(3) (6)  /  (3)
    (4)       (3)                  (4) (3)            (5) (4)     (5) (4)     (5) (4)
 Diagonal cannot be drawn   Diagonal cannot be drawn
    Number of cases = 1       Number of cases = 1         Number of cases = 3
		

Links

• Muhammed Sefa Saydam, Table of n, a(n) for n = 3..104

Crossrefs

Cf. A004149, A093128.

Programs

PARI

seq(n) = my(g=2/(1 - x + x^2 + x^3 + sqrt((1-x^4)*(1-2*x-x^2) + O(x*x^n)))); Vec((1 - x^2 - 2*x^3)*g - 1 - x + 2*x^3 + 2*x^4, -n+2) \\ Andrew Howroyd, Aug 28 2025

Formula

a(n) = A004149(n) - A004149(n-2) - 2*A004149(n-3) for n >= 5.

G.f.: (1 - x^2 - 2*x^3)*B(x) - 1 - x + 2*x^3 + 2*x^4, where B(x) is the g.f. of A004149. - Andrew Howroyd, Aug 28 2025