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Robert A. Russell, <a href="/A047773/a047773.txt">Mathematica Graphics3D program for A047773 examples</a>

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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 KD 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

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

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 K 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

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

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 *)

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

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(PARI) /* here U=A047749, V=A047750, K=A0047751A047751, and Q=A0047764A047764 */