A369315 Number of chiral pairs of polyominoes composed of n square cells of the hyperbolic regular tiling with Schläfli symbol {4,oo}.

2, 9, 48, 231, 1188, 6114, 32448, 175032, 962472, 5370524, 30377504, 173816313, 1004823816, 5861490300, 34468767840, 204161269620, 1217143807770, 7299003615537, 44005594027200, 266608363362900

Offset

4,1

Comments

A stereographic projection of the {4,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..23.

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

Formula

a(n) = A005034(n) - A005036(n) = (A005034(n) - A047749(n)) / 2 = A005036(n) - A047749(n).

Example

 __ __ __    __ __ __     __ __          __ __
|__|__|__|  |__|__|__|   |__|__|__    __|__|__|  a(4) = 2.
      |__|  |__|            |__|__|  |__|__|

Mathematica

p=4; 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: A005034 (oriented), A005036 (unoriented), A047749 (achiral), A369314 {3,oo}.

Sequence in context: A378074 A289576 A370200 * A223832 A323958 A306356

Adjacent sequences: A369312 A369313 A369314 * A369316 A369317 A369318

Keyword

easy,nonn

Author

Robert A. Russell, Jan 19 2024

Status

approved