A348096 - OEIS

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

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OFFSET

1,10

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.

LINKS

Table of n, a(n) for n=1..44.

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.

EXAMPLE

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).

Sequence in context: A187557 A270388 A052420 * A162971 A360829 A078521

Adjacent sequences: A348093 A348094 A348095 * A348097 A348098 A348099

KEYWORD

tabf,nonn,more

AUTHOR

R. J. Mathar, Sep 30 2021

EXTENSIONS

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

STATUS

approved