A371351 -id:A371351 - cp's OEIS frontend

A047773 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type D.

0, 0, 0, 1, 0, 0, 1, 1, 0, 2, 3, 0, 3, 5, 0, 7, 11, 0, 12, 23, 0, 30, 55, 0, 55, 114, 0, 143, 272, 0, 273, 588, 0, 728, 1428, 0, 1428, 3156, 0, 3876, 7750, 0, 7752, 17427, 0, 21318, 43263, 0, 43263, 98516, 0, 120175, 246672, 0, 246675, 567281, 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 D achiral symmetry and n tetrahedral cells. The center of symmetry is the altitude of a tetrahedral cell (32); the order of the symmetry group is 6. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 23 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 A047773 examples

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047749 (type U), A047750 (type V), A047751 (type K), A047764 (type Q).

Table[Switch[Mod[n,6],1,If[1==n,0,3Binomial[(n-1)/2,(n-1)/6]/(n+2)],2,6Binomial[n/2,(n-2)/6]/(n+4)-3Binomial[(n-2)/2,(n-2)/6]/(2n+2)-If[2==Mod[n,12],3Binomial[(n-2)/4,(n-2)/12],6Binomial[(n-4)/4,(n-8)/12]]/(n+4),4,6Binomial[(n-2)/2,(n-4)/6]/(n+2),5,3Binomial[(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,60}] (* Robert A. Russell, Mar 23 2024 *)

/* here U=A047749, V=A047750, K=A047751, and Q=A047764 */
U(n)={if(n%2,my(m=(n-1)/2);(3*m+1)!/((m+1)!*(2*m+1)!),my(m=n/2);(3*m)!/(m!*(2*m+1)!))};
V(n)={if(n%2,my(m=(n-1)/2);6*(3*m+2)!/(m!*(2*m+3)!),my(m=n/2);(3*m)!*(5*m+1)/((m+1)!*(2*m+1)!))};
K(n)={if(n==1,1,if(n<5,0,if(n%12==5,U((n-5)/12),0)))};
Q(n)={if(n<8,0,if(n%6==2,U((n-2)/6),0))};
D(n)={if(n<3||n%3==0,0,if(n%3==1,U((n-1)/3),(1/2)*(V((n-2)/3)-2*K(n)-Q(n))))};
for(k=1,57,print1(D(k),", ")) \\ Hugo Pfoertner, Mar 07 2020

If n=3m+2 then (1/2)*(A047750(m) - 2*A047751(n) - A047764(n)), if n=3m+1 then A047749(m), otherwise 0.

G.f.: (G(z^6)-1)/z + z*G(z^6) - z + z^2*G(z^6)^2 + z^4*G(z^6)^2 - z^5*G(z^24) - z^17*G(z^24)^2 - (z^2*G(z^6) + 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 23 2024

A047754 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type H.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 26, 0, 0, 0, 133, 0, 0, 0, 708, 0, 0, 0, 3861, 0, 0, 0, 21604, 0, 0, 0, 123266, 0, 0, 0, 715221, 0, 0, 0, 4206956, 0, 0, 0, 25032840, 0, 0, 0, 150413348, 0, 0, 0, 911379384, 0, 0, 0, 5562367173, 0, 0, 0, 34164355848

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 H achiral symmetry and n tetrahedral cells. The axis of symmetry is the line joining the centers of opposite edges of a tetrahedral cell (31); the order of the symmetry group is 4. 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 A047754 example

Cf. A047755.

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047751 (type K), A047753 (type I).

Table[If[1==Mod[n,4],Binomial[(3n-3)/4,(n-1)/4]/(n+1)-If[1==Mod[n,8],2Binomial[(3n-3)/8,(n-1)/8],4Binomial[(3n-7)/8,(n-5)/8]]/(n+3),0],{n,40}] (* Robert A. Russell, Mar 22 2024 *)

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

G.f.: z * (G(z^4) - G(z^8) - z^4*G(z^8)^2) / 2, 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

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A047773 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type D.

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A047754 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type H.

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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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