A348096 Array A(n,s) read by rows: the free n-polysticks of the square lattice with symmetry group of order 2^s.

0, 0, 1, 0, 0, 1, 1, 0, 1, 3, 1, 0, 8, 5, 1, 2, 39, 14, 2, 0, 187, 31, 4, 0, 880, 66, 4, 0, 4109, 142, 12, 2, 19274, 310, 7, 0, 90965, 694, 19, 0, 432545, 1445, 15, 0

Offset: 1

Author

R. J. Mathar, Sep 30 2021

Keywords

• tabf
• nonn
• more

Comments

The array has 4 columns for symmetry groups of order 1, 2, 4 and 8 (subgroups of D_8).

Polysticks with group order 1 have no symmetry. Polysticks with group order 2 have either a mirror line (parallel to edges or along a diagonal of the lattice) or a rotation axis of order 2 (180-degree rotation). Polysticks of group order 4 have two orthogonal mirror lines and the 180-degree rotation. Polysticks of group order 8 have in addition a rotation axis or order 4 (90-degree rotations), i.e. the full symmetry of the square.

Examples

			The array starts
      0   0  1 0
      0   1  1 0
      1   3  1 0
      8   5  1 2
     39  14  2 0
    187  31  4 0
    880  66  4 0
   4109 142 12 2
  19274 310  7 0
  90965 694 19 0
A(4,3)=2 counts the fully-symmetric unit square and the cross.
		

Crossrefs

Cf. A019988 (row sums), A096267 (fixed polysticks).

Formula

Sum_{s=0..3} A(n,s) = A019988(n).

8*A(n,0) + 4*A(n,1) + 2*A(n,2) + A(n,3) = A096267(n).

A(n,3) = 0 if n is not a multiple of 4.

Extensions

Row n=11 added.- R. J. Mathar, Oct 05 2021