A369471 Number of chiral pairs of polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}.

4, 24, 172, 1144, 8056, 57800, 427006, 3221216, 24773668, 193592840, 1534006620, 12301987920, 99699269740, 815520435048, 6725987757744, 55882659600320, 467387108739408, 3932600291539096, 33269691987278258, 282863688830816184

Offset: 4

Author

Robert A. Russell, Jan 23 2024

Keywords

• easy
• nonn

Comments

A stereographic projection of the {5,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

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

Crossrefs

Polyominoes: A005038 (oriented), A005040 (unoriented), A369472 (achiral), A369315 {4,oo}.

Programs

Mathematica

p=5; 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}]

Formula

a(n) = A005038(n) - A005040(n) = (A005038(n) - A369472(n)) / 2 = A005040(n) - A369472(n).