OFFSET
1,5
COMMENTS
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
LINKS
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
FORMULA
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
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved