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

0,2

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.

Links

Table of n, a(n) for n=0..17.

Formula

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

Example

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.

Mathematica

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

Prog

(Sage)
def A367872(n):
    return binomial(5*n+2, n) * (n+3) / (4*n+3)
(PARI) for(n=0, 25, print1(binomial(5*n+2, n)*(n+3)/(4*n+3), ", "))

Crossrefs

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

Sequence in context: A340895 A220442 A215698 * A179540 A370346 A375172

Adjacent sequences: A367869 A367870 A367871 * A367873 A367874 A367875

Keyword

nonn

Author

F. Chapoton, Feb 22 2024

Status

approved