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A119611 Number of free polyominoes in {4,5} tessellation of the hyperbolic plane. 0
1, 1, 1, 2, 5, 15, 56 (list; graph; refs; listen; history; text; internal format)



Each tessellation in the hyperbolic plane is represented by a Schläfli symbol of the form {p,q}, which means that q regular p-gons surround each vertex. There exists a hyperbolic tessellation {p,q} for every p,q such that (p-2)*(q-2) > 4. I have, in a paper not referenced here explicitly, described polyiamonds and enumerated in the Klein curve, topologically derived from the {3,7} or dually the {7,3} hyperbolic tessellation, as a side-effect of defining polyheptagons in the Klein quartic.


Table of n, a(n) for n=0..6.

Don Hatch, Hyperbolic Planar Tesselations: {4,5}.

Eric Weisstein's World of Mathematics, Polyomino.


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

For n = 4, the square tetromino does not exist. In its place is the cut-square, a pentagonal pentomino with one cell removed; see n = 5.

For n = 5, we have modified versions of 11 of the 12 Euclidean pentominoes, but not the P-pentomino, as that has the square tetromino as a subpolyomino, with one adjacent monomino. In place of the P we have 4 unique hyperbolic pentominoes. First, the aforementioned pentagonal pentomino, with 5-fold symmetry, embedded in the space where 5 right angles define a full rotation. Next, the cut-square tetromino can have an adjacent monomino in 4 nonisomorphic positions. 12 - 1 + 4 = 15 hyperbolic pentominoes.

For n = 6 we lose the 8 Euclidean hexominoes that have the square tetromino as a subpolyomino. In their place, we have the pentagonal pentomino with an adjacent monomino; the cut-square with an adjacent domino in 12 nonisomorphic positions; and the cut-square with two separate adjacent monominoes in 16 nonisomorphic positions. 35 - 8 + 29 = 56 hyperbolic hexominoes.

Comment about the example "For = 5, etc.": What? You have described five replacements for the P. So I count 12 - 1 + 5 = 16. - Don Knuth, Oct 14 2016


Cf. A000105.

Sequence in context: A137533 A121392 A216388 * A005976 A187981 A048192

Adjacent sequences:  A119608 A119609 A119610 * A119612 A119613 A119614




Jonathan Vos Post, Jun 04 2006



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Last modified February 23 02:33 EST 2020. Contains 332159 sequences. (Running on oeis4.)