A005034 -id:A005034 - cp's OEIS frontend

A385149 Number of chiral pairs of asymmetric polyominoes with n cells of the regular tiling with Schläfli symbol {4,oo}.

0, 0, 0, 0, 1, 8, 43, 225, 1162, 6081, 32315, 174856, 961764, 5369567, 30373643, 173811011, 1004802212, 5861460314, 34468644574, 204161097084, 1217143092549, 7299002607829, 44005589820244, 266608357403244, 1622502342468552, 9914884364399700

Offset: 0

Robert A. Russell, Jun 19 2025

A stereographic projection of the {4,oo} tiling on the Poincaré disk can be obtained via the Christensson link. Each member of a chiral pair is a reflection but not a rotation of the other.

			 __ __ __    __ __ __
|__|__|__|  |__|__|__|  a(4) = 1.
      |__|  |__|

Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
Robert A. Russell, Stereographic projections of chiral pairs of asymmetric polyominoes with 4 or 5 cells

Cf. A005034 (oriented), A005036 (unoriented), A369315 (chiral), A047749 (achiral), A001764 (rooted).

Table[If[n<4,0,(3Binomial[3n,n]/(2n+1)-Binomial[3n+1,n]/(n+1) + Switch[Mod[n,4], 0,4Binomial[3n/4,n/4]/(n/2+1)-6Binomial[3n/2,n/2]/(n+1), 1,(4Binomial[(3n-3)/4,(n-1)/4]-10Binomial[(3n-1)/2,(n-1)/2])/(n+1)+(8Binomial[(3n+1)/4,(n-1)/4]+16Binomial[(3n-3)/4,(n-5)/4])/(n+3), 2,16Binomial[(3n-2)/4,(n-2)/4]/(n+2)-6Binomial[3n/2,n/2]/(n+1), 3,24Binomial[(3n-1)/4,(n-3)/4]/(n+3)-10Binomial[(3n-1)/2,(n-1)/2]/(n+1)])/8],{n,0,30}]

G.f.: (3*G(z) - G(z)^2 - 6*G(z^2) - 5z*G(z^2)^2 + 4*G(z^4) + 2z*G(z^4) + 2z*G(z^4)^2 + 4z^2*G(z^4)^2 + 4z^3*G(z^4)^3 + 2z^5*G(z^4)^4) / 8, where G(z)=1+z*G(z)^3 is the g.f. for A001764.

A296533 Number of nonequivalent noncrossing trees with n edges up to rotation and reflection.

Original entry on oeis.org

1, 1, 1, 3, 7, 28, 108, 507, 2431, 12441, 65169, 351156, 1926372, 10746856, 60762760, 347664603, 2009690895, 11723160835, 68937782355, 408323575275, 2434289046255, 14598013278960, 88011196469040, 533216762488020, 3245004785069892, 19829769013792908

Offset: 0

Andrew Howroyd, Dec 14 2017

nonn

The number of all noncrossing trees with n edges is given by A001764.

The number of nodes will be n + 1.

			Case n=3:
   o---o   o---o   o---o
   |       | \       \
   o---o   o   o   o---o
In total there are 3 distinct noncrossing trees up to rotation and reflection.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Bernold Fiedler, Real-time blow-up and connection graphs of rational vector fields on the Riemann sphere, arXiv:2504.20503 [math.DS], 2025. See p. 24.
Marc Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

Cf. A001764, A005034, A006013, A296532 (up to rotation only).

a[n_] := (If[OddQ[n], 3*Binomial[(1/2)*(3*n - 1), (n - 1)/2], Binomial[3*n/2, n/2]] + Binomial[3*n, n]/(2*n + 1))/(2*(n + 1));
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 27 2017, after Andrew Howroyd *)

a(n)={(binomial(3*n, n)/(2*n+1) + if(n%2, 3*binomial((3*n-1)/2, (n-1)/2),  binomial(3*n/2, n/2)))/(2*(n+1))}

a(2n) = (A296532(2n) + A001764(n))/2, a(2n-1) = (A296532(2n-1) + A006013(n-1))/2.

a(2n) = A005034(2n).

A317184 Number of "non-connected" chord diagrams of degree n.

Original entry on oeis.org

0, 1, 3, 12, 74, 647, 6961, 89739, 1337152, 22609111, 427604861, 8945156182, 205080060435, 5113424483894, 137759755603055, 3987742282544259, 123430664010624370, 4067922952436728611, 142218912116692593949

Offset: 1

N. J. A. Sloane, Jul 26 2018

nonn

Alexander Stoimenow, On the number of chord diagrams, Discr. Math. 218 (2000), 209-233.

Cf. A005034, A018225.

Stoimenow states that a Mma package is available from his website.

More terms from Sean A. Irvine, Feb 04 2019

cp's OEIS Frontend

A385149 Number of chiral pairs of asymmetric polyominoes with n cells of the regular tiling with Schläfli symbol {4,oo}.

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A296533 Number of nonequivalent noncrossing trees with n edges up to rotation and reflection.

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A317184 Number of "non-connected" chord diagrams of degree n.

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