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A068381 Number of partitions of n X n checkerboard by two edgewise-connected sets which produce the maximum n^2-2n+2 frontier edges between the two sets. 3
12, 32, 96, 648, 7736, 228424, 11974112, 1599762776, 382467306272, 234367651907856, 258981528765867728, 733498025032488425464, 3770347483688546402804760, 49588653272896250824990166768 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

Not divided by 4 because that property may not continue.

Each partition is counted twice in this sequence. The sequence can be computed by counting Hamiltonian paths on a n-1 x n-1 grid that start at any vertex on the grid boundary and terminate at another boundary vertex. Counts for whether the path starts or terminates on a corner or non-corner need to be computed separately as there are different multiplication factors. - Andrew Howroyd, Apr 13 2016

LINKS

Table of n, a(n) for n=2..15.

EXAMPLE

Illustration of a(2)=6*2:

    __.__     __.__     __.__    __.__     __.__     __.__

   |__|  |   |  |__|   |   __|  |__   |   |__.__|   |  |  |

   |__.__|   |__.__|   |__|__|  |__|__|   |__.__|   |__|__|

Illustration of relation of a Hamiltonian path in a 3 x 3 grid to solutions of a(4):

                 .__.__.__.__.   .__.__.__.__.   .__.__.__.__.   .__.__.__.__.

   .__.__        |__.__.__.  |   |  |__.__.  |   |__.__.__.  |   |  |__.__.  |

    __.__|  <=>  |  .__.__|  |   |  .__.__|  |   |  .__.__|  |   |  .__.__|  |

   |__.__.       |  |__.__.__|   |  |__.__.__|   |  |__.__.  |   |  |__.__.  |

                 |__.__.__.__|   |__.__.__.__|   |__.__.__|__|   |__.__.__|__|

CROSSREFS

Cf. A068392, A068393.

Cf. A001184, A000532, A121789.

Sequence in context: A243027 A242543 A194644 * A143238 A102091 A303079

Adjacent sequences:  A068378 A068379 A068380 * A068382 A068383 A068384

KEYWORD

nonn

AUTHOR

R. H. Hardin, Mar 04 2002

EXTENSIONS

a(7)-a(15) from Andrew Howroyd, Apr 13 2016

STATUS

approved

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Last modified December 10 04:15 EST 2019. Contains 329885 sequences. (Running on oeis4.)