Robert A. Russell - 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.

A371397 Number of chiral pairs of polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}.

Original entry on oeis.org

0, 0, 0, 1, 6, 54, 416, 3111, 22898, 168460, 1242985, 9227333, 68949103, 518618196, 3925228596, 29879207817, 228630283775, 1757699977107, 13570824097968, 105182547181534, 818093724437992, 6383353614308209

Offset: 1

Robert A. Russell, Mar 21 2024

Also called polycubes. Each member of a chiral pair is a reflection but not a rotation of the other.

			Polyominoes with cell centers at (0,0,0), (0,0,1), (0,1,1), (1,1,1) and (0,0,0), (0,1,0), (0,1,1), (1,1,1) are a chiral pair.

Cf. A000162 (oriented), A038119 (unoriented), A007743 (achiral), A001931 (fixed).

Component symmetries: A376964, A376965, A376966, A376967, A376968, A376975, A376976, A376982, A377127, A377128, A377131.

a(n) = A000162(n) - A038119(n) = (A000162(n) - A007743(n))/2 = A038119(n) - A007743(n).

a(17)-a(22) from John Mason, Sep 19 2024

A371351 Number of achiral polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 15, 37, 73, 182, 364, 952, 1944, 5169, 10659, 28842, 60115, 164450, 345345, 953814, 2016144, 5609760, 11920740, 33378072, 71250060, 200553733, 429757960, 1215177680, 2612635888, 7416503776

Offset: 1

Robert A. Russell, Mar 19 2024

nonn

Also number of achiral simplicial 3-clusters or stack polytopes with n tetrahedral cells. An achiral polyomino is identical to its reflection.

L. W. Beineke and R. E. Pippert Enumerating dissectable polyhedra by their automorphism groups, Can. J. Math., 26 (1974), 50-67
F. Hering et al., The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.

Sum of achiral symmetry types (A047775, A047773, A047760, A047754, A047753, A047751, A047771, A047766 [type N], A047765, A047764) in Beineke link.

Cf. A007173 (oriented), A027610 (oriented), A371350 (chiral), A001764 (rooted), A208355(n-1) {3,oo}, A182299 {3,3,3,oo}.

Table[(If[OddQ[n],3Binomial[(3n-1)/2,n],2Binomial[3n/2,n]]+If[1==Mod[n,4],3Binomial[(3n-3)/4,(n-1)/2],0]+If[2==Mod[n,6],3Binomial[n/2-1,(n-2)/3],0])/(3n+3),{n,30}]

a(n) = ([0==n mod 2]*2*C(3n/2,n) + [1==n mod 2]*3*C((3n-1)/2,n) + [1==n mod4]*3*C((3n-3)/4,(n-1)/2) + [2==n mod6]*3*C(n/2-1,(n-2)/3)) / (3n+3).
a(n) = 2*A027610(n) - A007173(n) = A007173(n) - 2*A371350(n) = A027610(n) - A371350(n).
a(n) = 2*H(3,n) - h(3,n) in Table 8 of Hering link.
G.f.: (-4 + 4*G(z^2) + 3z*G(z^2)^2 + 3z*G(z^4) + 2z^2*G(z^6)) / 6, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764.

A371350 Number of chiral pairs of polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}.

Original entry on oeis.org

0, 0, 0, 1, 3, 16, 78, 397, 2037, 10820, 58349, 320824, 1790189, 10125858, 57938771, 334941363, 1953830203, 11489589280, 68053757016, 405714603234, 2433001205088, 14668531344984, 88869454457853, 540834122500464

Offset: 1

Robert A. Russell, Mar 19 2024

nonn

Also number of chiral pairs of simplicial 3-clusters or stack polytopes with n tetrahedral cells. Each member of a chiral pair is a reflection but not a rotation of the other.

L. W. Beineke and R. E. Pippert Enumerating dissectable polyhedra by their automorphism groups, Can. J. Math., 26 (1974), 50-67
F. Hering et al., The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.

Sum of chiral symmetry types (A047776, A047774, A047762, A047758, A047752, A047769, A047766 [type O]) in Beineke article.

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A369314 {3,oo}, A369474 {3,3,3,oo}.

Table[Switch[Mod[n,3],1,Binomial[n,(n-1)/3],2,Binomial[n,(n-2)/3],_,0]/(3n)+(Binomial[3n,n]/(6n+3)-If[OddQ[n],Binomial[3(n-1)/2+1,n],Binomial[3n/2,n]/3]-2If[1==Mod[n,4],Binomial[(3n-3)/4,(n-1)/2],0]-2If[2==Mod[n,6],Binomial[n/2-1,n/3-2/3],0])/(4n+4),{n,30}]

a(n) = A007173(n) - A027610(n) = (A007173(n) - A371351(n))/2 = A027610(n) - A371351(n).
a(n) = h(3,n) - H(3,n) in Table 8 of Hering link.
G.f.: (4*G(z) - 2*G(z)^2 + z*G(z)^4 - 2*G(z^2) - 3z*G(z^2)^2 + 2z*(4 G(z^3) + 2z*G(z^3)^2 - 3*G(z^4) - 2z*G(z^6))) / 24.

A369474 Number of chiral pairs of polyominoes composed of n pentachoral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,3,oo}.

Original entry on oeis.org

0, 0, 0, 0, 1, 10, 80, 611, 4602, 34791, 265606, 2054034, 16094883, 127693729, 1024649237, 8306343347, 67952829212, 560471786912, 4656785469564, 38948533963500, 327715193729107, 2772468576820531

Offset: 1

Robert A. Russell, Mar 20 2024

nonn

Also number of chiral pairs of simplicial 4-clusters or stack polytopes with n pentachoral cells. Each member of a chiral pair is a reflection but not a rotation of the other. Some of the h(4,n) terms in the Hering article are in error, including the 6th, 8th and 9th.

F. Hering et al., The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.

Cf. A007175 (oriented), A182322 (oriented), A182299 (achiral), A002293 (rooted), A371350 {3,3,oo}.

This is the half the difference of A007175 and A182299, both of which have Mathematica programs.

a(n) = A007175(n) - A182322(n) = (A007175(n) - A182299(n))/2 = A182322(n) - A182299(n).

a(n) = h(4,n) - H(4,n) in Table 8 of Hering link.

A369473 Number of chiral pairs of polyominoes composed of n hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,oo}.

Original entry on oeis.org

7, 50, 448, 3810, 34200, 314655, 2982040, 28897440, 285577500, 2868769045, 29227672960, 301429078080, 3141983233130, 33059729519325, 350763428176480, 3749420512083472, 40348040467611800, 436827334389425980

Offset: 4

Robert A. Russell, Jan 23 2024

A stereographic projection of the {6,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.

Malin Christensson, Make hyperbolic tilings of images, web page, 2019.

Polyominoes: A221184(n-1) (oriented), A004127 (unoriented), A143546 (achiral), A369471 {5,oo}.

p=6; 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)-Binomial[((p-1)n+1)/2, (n-1)/2]/((p-1)n+1)], Binomial[(p-1)n/2, n/2]/((p-2)n+2)]+DivisorSum[GCD[p, n-1], EulerPhi[#]Binomial[((p-1)n+1)/#, (n-1)/#]/((p-1)n+1)&, #>1&])/2, {n, 4, 30}]

a(n) = A221184(n-1) - A004127(n) = (A221184(n-1) - A143546(n)) / 2 = A004127(n) - A143546(n).

A369471 Number of chiral pairs of polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}.

Original entry on oeis.org

4, 24, 172, 1144, 8056, 57800, 427006, 3221216, 24773668, 193592840, 1534006620, 12301987920, 99699269740, 815520435048, 6725987757744, 55882659600320, 467387108739408, 3932600291539096, 33269691987278258, 282863688830816184

Offset: 4

Robert A. Russell, Jan 23 2024

A stereographic projection of the {5,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.

Malin Christensson, Make hyperbolic tilings of images, web page, 2019.

Polyominoes: A005038 (oriented), A005040 (unoriented), A369472 (achiral), A369315 {4,oo}.

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)-Binomial[((p-1)n+1)/2, (n-1)/2]/((p-1)n+1)], Binomial[(p-1)n/2, n/2]/((p-2)n+2)]+DivisorSum[GCD[p, n-1], EulerPhi[#]Binomial[((p-1)n+1)/#, (n-1)/#]/((p-1)n+1)&, #>1&])/2, {n, 4, 30}]

a(n) = A005038(n) - A005040(n) = (A005038(n) - A369472(n)) / 2 = A005040(n) - A369472(n).

A369472 Number of achiral polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}.

Original entry on oeis.org

1, 1, 2, 4, 9, 22, 52, 140, 340, 969, 2394, 7084, 17710, 53820, 135720, 420732, 1068012, 3362260, 8579560, 27343888, 70068713, 225568798, 580034052, 1882933364, 4855986044, 15875338990, 41043559340, 134993766600

Offset: 1

Robert A. Russell, Jan 23 2024

A stereographic projection of the {5,oo} tiling on the Poincaré disk can be obtained via the Christensson link.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.

Column k=5 of A370060.

Polyominoes: A005038 (oriented), A005040 (unoriented), A369471 (chiral), A002293 (rooted), A047749 {4,oo}, A143546 {6,oo}.

p=5; Table[If[EvenQ[n],Binomial[(p-1)n/2,n/2]/((p-2)n/2+1),If[OddQ[p],(p-1)Binomial[(p-1)n/2-1,(n-1)/2]/((p-2)n+1),p Binomial[(p-1)n/2-1/2,(n-1)/2]/((p-2)n+2)]],{n,35}]

For n even, a(n) = C(2n,n/2)/(3n/2+1).
For n odd, a(n) = 4*C(2n-1,(n-1)/2)/(3n+1).
a(n+2)/a(n) ~ 256/27. a(2m+1)/a(2m) ~ 32/9; a(2m)/a(2m-1) ~ 8/3.
a(n) = 2*A005040(n) - A005038(n) = A005038(n) - 2*A369471(n) = A005040(n) - A369471(n).
G.f.: G(z^2)+z*G(z^2)^2, where G(z)=1+z*G(z)^4, the generating function for A002293.
a(2m) = A002293(m) ~ (4^4/3^3)^m*sqrt(4/(2*Pi*(3*m)^3)). - Robert A. Russell, Jul 15 2024

A369315 Number of chiral pairs of polyominoes composed of n square cells of the hyperbolic regular tiling with Schläfli symbol {4,oo}.

Original entry on oeis.org

2, 9, 48, 231, 1188, 6114, 32448, 175032, 962472, 5370524, 30377504, 173816313, 1004823816, 5861490300, 34468767840, 204161269620, 1217143807770, 7299003615537, 44005594027200, 266608363362900

Offset: 4

Robert A. Russell, Jan 19 2024

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) = 2.
      |__|  |__|            |__|__|  |__|__|

Malin Christensson, Make hyperbolic tilings of images, web page, 2019.

Polyominoes: A005034 (oriented), A005036 (unoriented), A047749 (achiral), A385149 (asymmetric), A001764 (rooted), A369314 {3,oo}.

p=4; 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)-Binomial[((p-1)n+1)/2,(n-1)/2]/((p-1)n+1)],Binomial[(p-1)n/2,n/2]/((p-2)n+2)]+DivisorSum[GCD[p,n-1],EulerPhi[#]Binomial[((p-1)n+1)/#,(n-1)/#]/((p-1)n+1)&,#>1&])/2,{n,4,30}]
Table[(3Binomial[3n,n]/(2n+1)-Binomial[3n+1,n]/(n+1)-If[OddQ[n],6Binomial[(3n-1)/2,(n-1)/2]-If[1==Mod[n,4],4Binomial[(3n-3)/4,(n-1)/4],0],2Binomial[3n/2,n/2]]/(n+1))/8,{n,0,30}] (* Robert A. Russell, Jun 19 2025 *)

a(n) = A005034(n) - A005036(n) = (A005034(n) - A047749(n)) / 2 = A005036(n) - A047749(n).

G.f.: (3*G(z) - G(z)^2 - 2*G(z^2) - 3z*G(z^2)^2 + 2z*G(z^4)) / 8, where G(z)=1+z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Jun 19 2025

A369314 Number of chiral pairs of polyominoes composed of n triangular cells of the hyperbolic regular tiling with Schläfli symbol {3,oo}.

Original entry on oeis.org

1, 2, 7, 22, 68, 214, 691, 2240, 7396, 24702, 83469, 284928, 981814, 3410990, 11939752, 42075308, 149180356, 531866972, 1905872189, 6861162880, 24805796984, 90035940942, 327988261992, 1198853954688, 4395798528850

Offset: 4

Robert A. Russell, Jan 19 2024

A stereographic projection of the {3,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; a(5)=2.             \/            \/             \/    \/

Malin Christensson, Make hyperbolic tilings of images, web page, 2019.

Polyominoes: A001683(n+2) (oriented), A000207 (unoriented), A208355(n-1) (achiral).

Table[Binomial[2n,n]/(2(n+1)(n+2))-If[OddQ[n],Binomial[n,(n+1)/2]/n,Binomial[n,n/2]/(n+2)]/2+If[Divisible[n-1,3],Binomial[(2n+1)/3,(n-1)/3]/(2n+1),0],{n,4,20}]

a(n) = C(2n,2)/(2(n+1)(n+2)) - [2\(n+1)]*C(n,(n+1)/2)/(2n) - [2\n]*C(n,n/2)/(2n+4) + [3\(n-1)]*C((2n+1)/3,(n-1)/3)/(2n+1).

a(n) = A001683(n+2) - A000207(n) = (A001683(n+2) - A208355(n-1)) / 2 = A000207(n) - A208355(n-1).

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User: Robert A. Russell

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

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A371397 Number of chiral pairs of polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}.

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A371351 Number of achiral polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}.

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A371350 Number of chiral pairs of polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}.

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A369474 Number of chiral pairs of polyominoes composed of n pentachoral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,3,oo}.

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A369473 Number of chiral pairs of polyominoes composed of n hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,oo}.

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A369471 Number of chiral pairs of polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}.

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A369472 Number of achiral polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}.

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