A047772 -id:A047772 - cp's OEIS frontend

A047759 Duplicate of A047772.

0, 0, 0, 1, 8, 42, 232, 1277, 7183, 41041, 238315, 1402076, 8343804, 50136483

Offset: 0

dead

A047758 Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type G.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 8, 0, 0, 0, 42, 0, 0, 0, 232, 0, 0, 0, 1277, 0, 0, 0, 7183, 0, 0, 0, 41041, 0, 0, 0, 238315, 0, 0, 0, 1402076, 0, 0, 0, 8343804, 0, 0, 0, 50136483, 0, 0, 0, 303790544, 0, 0, 0, 1854115285, 0, 0, 0, 11388104153

Offset: 1

N. J. A. Sloane

nonn

One of 17 different symmetry types comprising A007173 and A027610 and one of 7 for A371350. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type G chiral symmetry and n tetrahedral cells. The axis of symmetry is a line connecting the centers of opposite edges of a tetrahedral cell (31); the order of the symmetry group is 4. Each member of a chiral pair is a reflection but not a rotation of the other. - Robert A. Russell, Mar 22 2024

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
Robert A. Russell, Mathematica Graphics3D program for A047758 example

Cf. A047772.

Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047751 (type K), A047752 (type J), A047753 (type I).

Table[If[1==Mod[n,4],(2Boole[1==n]+2Binomial[(3n-3)/4,(n-1)/4]/(n+1)-If[1==Mod[n,8],12Binomial[(3n-3)/8,(n-1)/8]/(n+3),12Binomial[(3n-7)/8,(n+3)/8]/(n-1)]-If[5==Mod[n,12],6Binomial[(n-5)/4,(n-5)/12]/(n+1)-If[5==Mod[n,24],36Binomial[(n-5)/8,(n-5)/12],72Binomial[(n-9)/8,(n-17)/24]]/(n+7),0])/6,0],{n,50}] (* Robert A. Russell, Mar 22 2024 *)

If n=4m+1 then (1/6)*(A001764(m) - 3*A047753(n) - 2*A047752(n) - A047751(n)), otherwise 0.

G.f.: z * (2 + G(z^4) - z^4*G(z^12) - 3 * (G(z^8) + z^4*G(z^8)^2 - z^4*G(z^24) - z^16*G(z^24)^2)) / 6, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 22 2024

A047771 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type L.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 8, 0, 42, 0, 232, 0, 1277, 0, 7183, 0, 41041, 0, 238315, 0, 1402076, 0, 8343804, 0, 50136483, 0, 303790544, 0, 1854115285, 0, 11388104153, 0, 70338364135, 0, 436605050440, 0, 2722153369473, 0, 17040017600925, 0

Offset: 1

N. J. A. Sloane

nonn

One of 17 different symmetry types comprising A007173 and A027610 and one of 10 for A371351. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type L achiral symmetry and n tetrahedral cells. The plane of symmetry is a tetrahedral face (210); the order of the symmetry group is 2. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 22 2024

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
Robert A. Russell, Mathematica Graphics3D program for A047771 example.

Cf. A047772.

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047764 (type Q), A047765 (type P), A047766 (type N).

Table[(If[OddQ[n],0,Binomial[3n/2,n/2]/(n+1)-6If[OddQ[n/2],2Binomial[(3n-2)/4,(n-2)/4],Binomial[3n/4,n/4]]/(n+2)]-3If[2==Mod[n,6],Binomial[3(n-2)/6,(n-2)/6]/(n+1)-If[2==Mod[n,12],6Binomial[3(n-2)/12,(n-2)/12],12Binomial[n/4-1,(n-8)/12]]/(n+4),0])/6,{n,30}] (* Robert A. Russell, Mar 22 2024 *)

If n=2m then (1/6)*(A001764(m) - 2*A047766(n) - 3*A047765(n) - A047764(n)), otherwise 0.

G.f.: (2 + G(z^2) - z^2*G(z^6)) / 6 - (G(z^4) + z^2*G(z^4)^2 - z^2*G(z^12) - z^8*G(z^12)^2) / 2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 30 2024

A047775 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type B.

Original entry on oeis.org

0, 0, 0, 0, 2, 5, 11, 25, 66, 131, 349, 708, 1911, 3856, 10604, 21597, 59961, 123266, 345060, 715198, 2015416, 4206926, 11919257, 25032840, 71246129, 150413234, 429750208, 911379241, 2612614298, 5562367173, 15991792731, 34164355260

Offset: 1

N. J. A. Sloane

nonn

One of 17 different symmetry types comprising A007173 and A027610 and one of 10 for A371351. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type B achiral symmetry and n tetrahedral cells. The plane of symmetry bisects a tetrahedral cell (321); the order of the symmetry group is 2. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 29 2024

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
S. J. Cyvin, Jianji Wang, J. Brunvoll, Shiming Cao, Ying Li, B. N. Cyvin, and Yugang Wang, Staggered conformers of alkanes: complete solution of the enumeration problem, J. Molec. Struct. 413-414 (1997), 227-239.
Robert A. Russell, Mathematica Graphics3D program for A047775 examples

Cf. A047772.

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047749 (type U), A047751 (type K), A047753 (type I), A047760 (type F), A047764 (type Q), A047765 (type P), A047773 (type D).

Table[If[n<5,0,If[OddQ[n],2Binomial[(3n-1)/2,(n-1)/2]/(n+1)+If[1==Mod[n,4],2Binomial[3(n-1)/4,(n-1)/4]/(n+1),0],Binomial[3n/2,n/2]/(n+1)-If[OddQ[n/2],4Binomial[(3n-2)/4,(n-2)/4],2Binomial[3n/4,n/4]]/(n+2)]/2+If[2==Mod[n,6],3Binomial[(n-2)/2,(n-2)/6]/(n+1)+If[2==Mod[n,12],6Binomial[(n-2)/4,(n-2)/12],12Binomial[n/4-1,(n-8)/12]]/(n+4),0]/2-Switch[Mod[n,4],1,4Binomial[(3n+1)/4,(n-1)/4],3,2Binomial[(3n+3)/4,(n+1)/4],,0]/(n+3)-Switch[Mod[n,6],1,3Binomial[(n-1)/2,(n-1)/6]/(n+2),2,6Binomial[n/2,(n-2)/6]/(n+4),4,6Binomial[(n-2)/2,(n-4)/6]/(n+2),,0]-If[5==Mod[n,6],3Binomial[(n+1)/2,(n+1)/6]/(n+4)-Switch[Mod[n,24],5,12Binomial[(n-5)/8,(n-5)/24],17,24Binomial[(n-9)/8,(n-17)/24],,0]/(n+7),0]],{n,50}] (* _Robert A. Russell, Mar 29 2024 *)

a(n) = (1/2)*(A047749(n) - 2*A047773(n) - 2*A047760(n) - A047753(n) - A047751(n) - A047764(n) - A047765(n)).

G.f.: (2 - G(z^4) - G(z^6))/z + (G(z^2) + z*G(z^2)^2 - G(z^4) + z*G(z^4) - z^2*G(z^4)^2 + z^2*G(z^6) + z^2*G(z^12) + z^8*G(z^12)^2) / 2 + z - z*G(z^4)^2 - z*G(z^6) - z^2*G(z^6)^2 - z^4*G(z^6)^2 + z^5*G(z^24) + z^17*G(z^24)^2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 29 2024

A121180 Alkane systems (see Cyvin reference for precise definition).

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 6, 26, 32, 133, 176, 708, 952, 3861, 5302, 21604, 29960

Offset: 1

N. J. A. Sloane, Aug 17 2006

Appears to be A047774 without every third term (all omitted terms are zeros). - Andrey Zabolotskiy, Jul 29 2023

S. J. Cyvin, Jianji Wang, J. Brunvoll, Shiming Cao, Ying Li, B. N. Cyvin, and Yugang Wang, Staggered conformers of alkanes: complete solution of the enumeration problem, J. Molec. Struct. 413-414 (1997), 227-239. See Table 5, column C_2.

Cf. other columns of Cyvin et al.'s Table 5: A027610 (spectral isomers), A007173 (stereoisomers), A047775 (C_s), A047772 (C_i), A047774 (C_3, apparently), A047767 (C_{2h}), A047761 (C_{2v}), A047773 (C_{3v}, apparently).

Cf. A121181, A121189.

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A047759 Duplicate of A047772.

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A047758 Number of chiral pairs of dissectable polyhedra with n tetrahedral cells and symmetry of type G.

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A047771 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type L.

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A047775 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type B.

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A121180 Alkane systems (see Cyvin reference for precise definition).

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