A389561 - OEIS

A389561

Number of oriented polyominoes with n heptagonal cells of the hyperbolic regular tiling with Schläfli symbol {7,oo}.

1, 1, 1, 3, 26, 203, 1989, 20254, 219388, 2459730, 28431861, 336492104, 4062243024, 49863336565, 620840444834, 7825644901656, 99706732152034, 1282437129428070, 16633836691938270, 217372667817157545, 2859855105687384246, 37855395250770282075, 503863784437367408822, 6740453109521768696400, 90588122128165301157648, 1222637207555319112594128

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

A stereographic projection of the {7,oo} tiling on the Poincaré disk can be obtained via the Christersson link. For oriented polyominoes, chiral pairs are counted as two.

LINKS

Table of n, a(n) for n=0..25.

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

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.: (12*G(z) - 5*G(z)^2 + 7*G(z^2) + 12*z*G(z^7))/14, where G(z) = 1 + z*G(z)^6 is the g.f. for A002295.

a(n) = A005419(n) + A389562(n) = 2*A005419(n) - A389563(n) = 2*A389562(n) + A389563(n).

a(n) ~ (6^6/5^5)^n * sqrt(3 / (Pi * 5^5 * n^5)).

MATHEMATICA

p=7; Table[(Binomial[(p-1)n, n]/((p-2)n+1)+If[OddQ[n], 0, Binomial[(p-1)n/2, n/2]] + DivisorSum[GCD[p, n-1], EulerPhi[#]Binomial[((p-1)n+1)/#-1, (n-1)/#]&, #>1&])/((p-2)n+2), {n, 0, 40}]

CROSSREFS

Column k=7 of A295224.

Cf. A005419 (unoriented), A389562 (chiral), A389563 (achiral), A389564 (asymmetric), A002295 (rooted), A221184{n-1} {6,oo}, A389936 {8,oo}.

Sequence in context: A355047 A228116 A091123 * A037790 A037671 A037797

Adjacent sequences: A389558 A389559 A389560 * A389562 A389563 A389564

KEYWORD

nonn

AUTHOR

Robert A. Russell, Oct 08 2025

STATUS

approved