A349328 Number of polyominoes with n cells and exactly one line of reflection symmetry, where that one line is parallel to the grid and passes through the center of at least one square.

0, 0, 0, 1, 2, 4, 9, 16, 38, 62, 147, 241, 564, 926, 2148, 3561, 8195, 13700, 31349, 52858, 120357, 204444, 463712, 792986, 1792582, 3083469, 6950579, 12018394, 27023509, 46943409, 105320716, 183715445, 411364068, 720236762, 1609836928, 2828102115

Offset

1,5

Links

John Mason, Table of n, a(n) for n = 1..60

Toshihiro Shirakawa, Enumeration of Polyominoes considering the symmetry, April 2012, pp. 3-4.

Toshihiro Shirakawa, Enumeration of Polyominoes up to Size N=59, arXiv:2510.22446 [math.CO], 2025.

Formula

For odd n, a(n) = A006746(n).

For even n, a(n) = A006746(n) - A349329(n/2).

a(n) = (A346799(n) - A346799(n/2) - 2*A351190(n) - A351127(n)) / 2. - John Mason, Nov 24 2025

Example

a(4) is 1 because of the tetromino:
   O
  OOO

Crossrefs

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A349329.

Cf. A351127, A351190, A346799.

Sequence in context: A282776 A283015 A264629 * A358904 A283059 A283081

Adjacent sequences: A349325 A349326 A349327 * A349329 A349330 A349331

Keyword

nonn

Author

John Mason, Nov 15 2021

Status

approved