A389194 Number of chiral pairs of asymmetric polyominoes with n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}.

0, 0, 0, 0, 2, 24, 162, 1144, 7986, 57800, 426526, 3221214, 24770126, 193592840, 1533979736, 12301987920, 99699059364, 815520435048, 6725986076784, 55882659600320, 467387095067464, 3932600291539026, 33269691874495056, 282863688830816184, 2415930982770209502, 20721058586117557304, 178409928539352955382

Offset

0,5

Comments

A stereographic projection of the {5,oo} tiling on the Poincaré disk can be obtained via the Christersson 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=0..26.

Malin Christersson, Make hyperbolic tilings of images, web page, 2015.

Robert A. Russell, Stereographic projections of chiral pairs of asymmetric polyominoes with 4 cells.

Formula

G.f.: (10 + 10*z + 8*G(z) - 3*G(z)^2 - 15*G(z^2) - 10*z*G(z^2)^2 + 10*z^2*G(z^4)^2 - 2*z*G(z^5) + 10*z^6*G(z^10)^2) / 20, where G(z) = 1+z*G(z)^4 is the g.f. for A002293.

Mathematica

Table[If[n<4, 0, (8Binomial[4n, n]/(3n+1)-6Binomial[4n+1, n]/(3n+2)-If[OddQ[n], 40Binomial[2n-1, (n-1)/2]/(3n+1), 30Binomial[2n, n/2]/(3n+2)-If[OddQ[n/2], 80Binomial[n-1, (n-2)/4]/(3n+2), 0]]-If[1==Mod[n, 5], (10Binomial[(4n-4)/5, (n-1)/5]-If[OddQ[(n-1)/5], 200Binomial[(2n-7)/5, (n-6)/10], 0])/(3n+2), 0])/20], {n, 0, 30}]

Crossrefs

Cf. A005038 (oriented), A005040 (unoriented), A369471 (chiral), A369472 (achiral), A002293 (rooted), A385149 {4,oo}, A389337 {6,oo}.

Sequence in context: A288443 A108476 A157053 * A279853 A052411 A073066

Adjacent sequences: A389191 A389192 A389193 * A389195 A389196 A389197

Keyword

nonn

Author

Robert A. Russell, Sep 25 2025

Status

approved