A389938 Number of chiral pairs of polyominoes with n octagonal cells of the hyperbolic regular tiling with Schläfli symbol {8,oo}.

14, 149, 1848, 22540, 289692, 3840193, 52443776, 733258526, 10458395948, 151683425390, 2231663272160, 33242247632440, 500542075950840, 7608837723266290, 116642747746011648, 1801638426173345760, 28016657948886941936, 438349148460892099123, 6896594435032808485000, 109055532806207278756920

Offset

4,1

Comments

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.

Frank Harary, Edgar M. Palmer and Ronald C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.

Formula

G.f.: (7*G(z) - 3*G(z)^2 - 4*G(z^2) - 7*z*G(z^2)^4 + 2*z*G(z^4)^2 + 4*z*G(z^8)) / 16, where G(z) = 1 + z*G(z)^7 is the g.f. for A002296.

a(n) = A389936(n) - A389937(n) = (A389936(n) - A143547(n)) / 2 = A389937(n) - A143547(n).

a(n) ~ (7^7/6^6)^n * sqrt(7/(8*Pi*(6*n)^5)).

Mathematica

p=8; Table[((Binomial[(p-1)n, n]/((p-2)n+1)+DivisorSum[GCD[p, n-1], EulerPhi[#]Binomial[((p-1)n+1)/#-1, (n-1)/#]&, #>2&])/((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/2, n/2]/((p-2)n+2)])/2, {n, 4, 40}]

Crossrefs

Cf. A389936 (oriented), A389937 (unoriented), A143547 (achiral), A389939 (asymmetric), A002296 (rooted), A389562 {7,oo}.

Sequence in context: A081184 A032343 A394863 * A222614 A019521 A009614

Adjacent sequences: A389935 A389936 A389937 * A389939 A389940 A389941

Keyword

nonn

Author

Robert A. Russell, Oct 21 2025

Status

approved