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approved
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Revision History for A005034
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Number of nonequivalent dissections of a polygon into n quadrilaterals by nonintersecting diagonals up to rotation.
(history; published version)
(history; published version)
STATUS
proposed
reviewed
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editing
proposed
COMMENTS
Number of oriented polyominoes composed of n square cells of the hyperbolic regular tiling with Schläfli symbol {4,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. For oriented polyominoes, chiral pairs are counted as two. - Robert A. Russell, Jan 19 20 2024
LINKS
Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
COMMENTS
Number of oriented polyominoes composed of n square cells of the hyperbolic regular tiling with Schläfli symbol {4,oo}. For oriented polyominoes, chiral pairs are counted as two. - Robert A. Russell, Jan 19 2024
FORMULA
MATHEMATICA
p=4; Table[Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) +If[OddQ[n], 0, Binomial[(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}]], {n, 1, 0, 20}] (* Robert A. Russell, Dec 11 2004 *)
CROSSREFS
STATUS
approved
editing
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proposed
approved
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editing
proposed
LINKS
Alexander Stoimenow, <a href="httphttps://www.math.torontodoi.edu/stoimenoorg/ncd.ps10.gz1016/S0012-365X(99)00347-7">On the number of chord diagrams</a>, Discr. Math. 218 (2000), 209-233. See p. 232.
STATUS
approved
editing
STATUS
reviewed
approved