A389937 Number of unoriented polyominoes with n octagonal cells of the hyperbolic regular tiling with Schläfli symbol {8,oo}.

1, 1, 1, 4, 21, 183, 1918, 22908, 290511, 3844688, 52454248, 733317578, 10458537726, 151684239896, 2231665269848, 33242259265880, 500542104940515, 7608837893841013, 116642748176333281, 1801638428726044480, 28016657955390294792, 438349148499724907709, 6896594435132535158130

Offset

0,4

Comments

For unoriented polyominoes, chiral pairs are counted as one.

Links

Table of n, a(n) for n=0..22.

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 + 12*G(z^2) + 9*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) - A389938(n) = (A389936(n) + A143547(n)) / 2 = A389938(n) + A143547(n).

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)], 3Binomial[(p-1)n/2, n/2]/((p-2)n+2)])/2, {n, 0, 40}]

Crossrefs

Column k=8 of A295260.

Cf. A389936 (oriented), A389938 (chiral), A143547 (achiral), A389939 (asymmetric), A002296 (rooted), A005419 {7,oo}.

Sequence in context: A393742 A393743 A393744 * A221370 A224500 A158108

Adjacent sequences: A389934 A389935 A389936 * A389938 A389939 A389940

Keyword

nonn

Author

Robert A. Russell, Oct 21 2025

Status

approved