%I M1851 #51 Jan 26 2024 08:36:08
%S 1,1,2,8,33,194,1196,8196,58140,427975,3223610,24780752,193610550,
%T 1534060440,12302123640,99699690472,815521503060,6725991120004,
%U 55882668179880,467387136083296,3932600361607809,33269692212847056,282863689410850236,2415930985594609548
%N Number of nonequivalent dissections of a polygon into n pentagons by nonintersecting diagonals up to rotation and reflection.
%C Number of unoriented polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. For unoriented polyominoes, chiral pairs are counted as one. - _Robert A. Russell_, Jan 23 2024
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A005040/b005040.txt">Table of n, a(n) for n = 1..1000</a>
%H Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%H F. Harary, E. M. Palmer, R. C. Read, <a href="/A000108/a000108_20.pdf">On the cell-growth problem for arbitrary polygons, computer printout, circa 1974</a>
%H F. Harary, E. M. Palmer and R. C. Read, <a href="http://dx.doi.org/10.1016/0012-365X(75)90041-2">On the cell-growth problem for arbitrary polygons</a>, Discr. Math. 11 (1975), 371-389.
%H E. V. Konstantinova, <a href="http://www.plouffe.fr/OEIS/citations/01-06.pdf">A survey of the cell-growth problem and some its variations</a>, preprint, 2001.
%H E. V. Konstantinova, <a href="http://com2mac.postech.ac.kr/">Com2Mac - Preprints</a> [Dead link?]
%F See Mathematica code.
%F a(n) ~ 2^(8*n - 1/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n + 5/2)). - _Vaclav Kotesovec_, Mar 13 2016
%F a(n) = A005038(n) - A369471(n) = (A005038(n) + A369472(n)) / 2 = A369471(n) + A369472(n). - _Robert A. Russell_, Jan 23 2024
%t 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)], 3Binomial[(p-1)n/2, n/2]/((p-2)n+2)]+Plus @@ Map[EulerPhi[ # ]Binomial[((p-1)n+1)/#, (n-1)/# ]/((p-1)n+1)&, Complement[Divisors[GCD[p, n-1]], {1, 2}]])/2, {n, 1, 20}] (* _Robert A. Russell_, Dec 11 2004 *)
%Y Column k=5 of A295260.
%Y Polyominoes: A005038 (oriented), A369471 (chiral), A369472 (achiral), A000207 {3,oo}, A005036 {4,oo}, A004127 {6,oo}, A005419 {7,oo}.
%K nonn
%O 1,3
%A _N. J. A. Sloane_
%E More terms from _Sascha Kurz_, Oct 13 2001.
%E Name edited by _Andrew Howroyd_, Nov 20 2017.