A047776 Number of chiral pairs of asymmetric dissectable polyhedra with n tetrahedral cells (type A).

0, 0, 0, 0, 2, 11, 71, 370, 2005, 10682, 58167, 320116, 1789210, 10121965, 57933469, 334919626, 1953800059, 11489466014, 68053583772, 405713887061, 2433000197471, 14668527134167, 88869448492895, 540834097467624, 3304961431043989, 20273201718862728, 124798671079300720, 770762029389852807

Offset

1,5

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 asymmetric (type A) with n tetrahedral cells. The order of the symmetry group is 1. Each member of a chiral pair is a reflection but not a rotation of the other. - Robert A. Russell, Mar 31 2024

Links

Table of n, a(n) for n=1..28.

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 A047776 examples

Formula

From Robert A. Russell, Mar 31 2024: (Start)

a(n) = A001764(n)/(12(n+1)) - A047775(n)/2 - A047774(n)/3 - A047773(n)/6 - A047762(n)/2 - A047760(n)/4 - A047758(n)/4 - A047754(n)/4 - A047753(n)/8 - A047752(n)/12 - A047751(n)/24 - A047771(n)/2 - A047769(n)/2 - A047766(n)/6 - A047766(n)/6 - A047765(n)/4 - A047764(n)/12.

G.f.: (G(z^4) + G(z^6) - 2)/(2z) - z/3 + G(z)/6 - G(z)^2/12 + z*G(z)^4/24 - 7*G(z^2)/12 - 3z*G(z^2)^2/8 - z*G(z^3)/6 - z^2*G(z^3)^2/12 + G(z^4)/2 - z*G(z^4)/6 + (z*G(z^4)^2 + z^2*G(z^4)^2 + z*G(z^6))/2 + z^2*G(z^6)/12 + (z^2*G(z^6)^2 + z^4*G(z^6)^2 - z^2*G(z^12))/2 + z^5*G(z^12)/6 - (z^8*G(z^12)^2 + z^5*G(z^24) + z^17*G(z^24)^2)/2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. (End)

Mathematica

Table[If[n < 5, 0, Binomial[3 n, 2 n + 2]/(3 n (n - 1))
    - If[OddQ[n], Binomial[3 n/2 - 1/2, n + 1] 3/(n - 1),
     7 Binomial[3 n/2, n + 1]/(3 n)]
    - Switch[Mod[n, 3], 1, Binomial[n - 1, 2 n/3 + 1/3]/(n - 1), 2,
     Binomial[n - 1, 2 n/3 + 2/3]/(n - 2), _, 0]
    + Switch[Mod[n, 4], 1, Binomial[3 n/4 - 3/4, n/2 + 1/2] 2/(3 (n - 1))
      + Binomial[3 n/4 + 1/4, n/2 + 3/2] 4/(n - 1) +
      Binomial[3 n/4 - 3/4, n/2 + 1/2] 4/(n + 3), 2,
     Binomial[3 n/4 - 1/2, n/2 + 1] 8/(n - 2), 3,
     Binomial[3 n/4 - 1/4, n/2 + 3/2] 12/(n - 3), 0,
     Binomial[3 n/4 - 1, n/2 + 1] 12/(n - 4)] +
    Switch[Mod[n, 6], 1, Binomial[n/2 - 1/2, n/3 + 2/3] 6/(n - 1), 2,
     Binomial[n/2 - 1, n/3 + 1/3] 4/(n - 2) +
      Binomial[n/2, n/3 + 4/3] 6/(n - 2) +
      Binomial[n/2 - 1, n/3 + 1/3] 6/(n + 4), 4,
     Binomial[n/2 - 1, n/3 + 2/3] 12/(n - 4), 5,
     Binomial[n/2 - 1/2, n/3 + 1/3] 9/(n + 4), _, 0] +
    Switch[Mod[n, 12], 2, -Binomial[n/4 - 1/2, n/6 + 2/3] 12/(n - 2), 5,
     Binomial[n/4 - 5/4, n/6 - 5/6] 2/(n + 1),
     8, -Binomial[n/4 - 1, n/6 - 1/3] 12/(n + 4), _, 0] -
    Switch[Mod[n, 24], 5, Binomial[n/8 - 5/8, n/12 - 5/12] 12/(n + 7), 17,
     Binomial[n/8 - 9/8, n/12 - 5/12] 24/(n + 7), _, 0]]/2, {n, 1, 60}] (* Robert A. Russell, Apr 09 2012 *)

Crossrefs

Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047775 (type B), A047774 (type C). A047773 (type D), A047762 (type E), A047760 (type F), A047758 (type G), A047754 (type H), A047753 (type I), A047752 (type J), A047751 (type K), A047771 (type L), A047769 (type M), A047766 (type N|O), A047765 (type P), A047764 (type Q).

Sequence in context: A250887 A371577 A334048 * A214692 A186633 A291301

Adjacent sequences: A047773 A047774 A047775 * A047777 A047778 A047779

Keyword

nonn,easy,changed

Author

N. J. A. Sloane

Status

approved