A358289 Generalized Gerrymander sequence: number of ordered ways to divide an n X n square into two connected regions, both of area n^2/2 if n is even, or of areas (n^2-1)/2 and (n^2+1)/2 if n is odd.

0, 4, 16, 140, 2804, 161036, 27803749, 14314228378, 21838347160809, 99704315229167288, 1367135978051264146578, 56578717186086829451888706, 7065692298178203128922479762418, 2670113158846160742372913777087464324, 3052313665715695874527667027409186333152556

Offset

1,2

Links

Anthony J. Guttmann and Iwan Jensen, Table of n, a(n) for n = 1..17

Anthony J. Guttmann and Iwan Jensen, The gerrymander sequence, or A348456, arXiv:2211.14482 [math.CO], 2022.

Formula

a(2*n)/2 = A348456(n).

Example

For n = 2, the square can be split vertically or horizontally, and then there are two ways to order the regions, so a(2) = 2*2 = 4.
For n = 3 we must choose a connected region of area 4 with a connected complement of area 5.
The possibilities are
  XXO      XXX      XXX
  XXO      XOO      OXO
  OOO      OOO      OOO
  4 ways   8 ways   4 ways
so a(3) = 16.

Crossrefs

Cf. A348456.

Sequence in context: A136651 A195899 A362143 * A156482 A335832 A173346

Adjacent sequences: A358286 A358287 A358288 * A358290 A358291 A358292

Keyword

nonn

Author

N. J. A. Sloane, Nov 25 2022

Status

approved