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Number of free polyominoes in {4,5} tessellation of the hyperbolic plane.
5

%I #70 Aug 25 2024 16:59:46

%S 1,1,1,2,5,16,55,224,978,4507,21430,104423,517897,2606185,13272978

%N Number of free polyominoes in {4,5} tessellation of the hyperbolic plane.

%H Code Golf Stack Exchange, <a href="https://codegolf.stackexchange.com/a/200295/53884">Impress Donald Knuth by counting polyominoes on the hyperbolic plane</a>.

%H Don Hatch, <a href="http://www.plunk.org/~hatch/HyperbolicTesselations/">Hyperbolic Planar Tesselations: {4,5}</a>.

%H Peter Kagey, <a href="/A119611/a119611.pdf">Example of the a(5)=16 free pentominoes in {4,5} tessellation of the hyperbolic plane</a>.

%H Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, <a href="https://doi.org/10.1007/s00026-022-00631-1">Extremal {p, q}-Animals</a>, Ann. Comb. (2023), p. 3.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polyomino.html">Polyomino</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Order-5_square_tiling">Order-5 square tiling</a>.

%e For n = 0,1,2,3 the polyominoes in the {4,5} tessellation of the hyperbolic plane are essentially same as the ordinary polyominoes in the plane (A000105), with redefinition of "straight line" and angular deficiency at a vertex.

%e For n = 4, the square tetromino does not exist. In its place is the cut-square, a pentagonal pentomino with one cell removed.

%e For n = 5, see links section.

%o (GAP) # See the Code Golf Stack Exchange link.

%o (bc) /* See the Code Golf Stack Exchange link. */

%o (C) // See the Code Golf Stack Exchange link.

%Y Cf. A000105.

%K nonn,more

%O 0,4

%A _Jonathan Vos Post_, Jun 04 2006

%E a(5) corrected by _Don Knuth_

%E a(6) corrected by _Christian Sievers_

%E a(7)-a(10) from _Christian Sievers_

%E a(11)-a(14) from _Ed Wynn_, Feb 14 2021