A369473 Number of chiral pairs of polyominoes composed of n hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,oo}.

7, 50, 448, 3810, 34200, 314655, 2982040, 28897440, 285577500, 2868769045, 29227672960, 301429078080, 3141983233130, 33059729519325, 350763428176480, 3749420512083472, 40348040467611800, 436827334389425980

Offset

4,1

Comments

A stereographic projection of the {6,oo} tiling on the Poincaré disk can be obtained via the Christensson link. Each member of a chiral pair is a reflection but not a rotation of the other.

Links

Table of n, a(n) for n=4..21.

Malin Christensson, Make hyperbolic tilings of images, web page, 2019.

Formula

a(n) = A221184(n-1) - A004127(n) = (A221184(n-1) - A143546(n)) / 2 = A004127(n) - A143546(n).

Mathematica

p=6; Table[(Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2))-If[OddQ[n], If[OddQ[p], Binomial[(p-1)n/2, (n-1)/2]/n, (p+1)Binomial[((p-1)n-1)/2, (n-1)/2]/((p-2)n+2)-Binomial[((p-1)n+1)/2, (n-1)/2]/((p-1)n+1)], Binomial[(p-1)n/2, n/2]/((p-2)n+2)]+DivisorSum[GCD[p, n-1], EulerPhi[#]Binomial[((p-1)n+1)/#, (n-1)/#]/((p-1)n+1)&, #>1&])/2, {n, 4, 30}]

Crossrefs

Polyominoes: A221184(n-1) (oriented), A004127 (unoriented), A143546 (achiral), A369471 {5,oo}.

Sequence in context: A367235 A355171 A266432 * A267243 A197570 A319884

Adjacent sequences: A369470 A369471 A369472 * A369474 A369475 A369476

Keyword

easy,nonn

Author

Robert A. Russell, Jan 23 2024

Status

approved