A367872 Number of dissections of a convex (4n+4)-sided polygon into n hexagons and one square (up to equivalence).

1, 4, 30, 272, 2695, 28080, 302064, 3321120, 37095201, 419276660, 4782798020, 54960207120, 635339153865, 7380876649216, 86101923008160, 1007980225327680, 11836181297108565, 139353762142502100

Offset: 0

Author

F. Chapoton, Feb 22 2024

Keywords

• nonn

Comments

This sequence counts dissections of a convex 4n+4-sided polygon into one square and n hexagons, modulo a simple equivalence relation. The equivalence relation is not defined by a group, but by local moves. Consider the octagon formed by a hexagon adjacent to the square. The local move is half-rotation of such octagons.

It seems that a(n) is divisible by n+1.

Examples

			For n=0, there is just one square, so that a(0)=1. For n=1, one can dissect an octagon in 8 ways into a hexagon and a square. In this case, the equivalence relation just relates every such dissection to its half rotated image, so that a(1)=4.
		

Crossrefs

Cf. A174687, A185113 (similar), A118970 (related).

Programs

Mathematica

Table[Binomial[5*n + 2, n]*(n + 3)/(4*n + 3), {n, 0, 50}]

PARI

for(n=0,25, print1(binomial(5*n+2,n)*(n+3)/(4*n+3), ", "))

Sage

def A367872(n):
    return binomial(5*n+2, n) * (n+3) / (4*n+3)

Formula

a(n) = binomial(5*n+2,n)*(n+3)/(4*n+3).