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

1,17

Comments

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

Links

Table of n, a(n) for n=1..65.

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

Formula

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

Mathematica

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 *)

Crossrefs

Cf. A047772.

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

Sequence in context: A028600 A299165 A221890 * A186980 A171918 A265117

Adjacent sequences: A047755 A047756 A047757 * A047759 A047760 A047761

Keyword

nonn

Author

N. J. A. Sloane

Status

approved