cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A068416 Number of partitionings of n X n checkerboard into two edgewise-connected sets.

Original entry on oeis.org

0, 6, 53, 627, 16213, 1123743, 221984391, 127561384993, 215767063451331, 1082828220389781579, 16209089366362071416785, 726438398002211876667379681, 97741115155002465272674416929195, 39565596445488219947994403962984729307
Offset: 1

Views

Author

R. H. Hardin, Mar 02 2002

Keywords

Comments

One of the partitions may completely surround the other. (Cf. A271802) - Andrew Howroyd, Apr 14 2016
Number of minimal edge cuts in the n X n grid graph. - Andrew Howroyd, Dec 11 2024

Examples

			Illustration of a(2)=6:
   11   12   12   12   11   11
   22   12   22   11   12   21
Illustration of a few solutions of a(3):
   111   112   122   111   111
   121   111   112   212   111
   111   111   222   222   222

Links

Crossrefs

Cf. A068392, A166755, A271802, A068381, A068417, A113900, A348456, A358289.

Formula

a(n) = A271802(n) + A140517(n-2). - Andrew Howroyd, Apr 14 2016
a(n) = A166755(n)/2. - Andrew Howroyd, Dec 11 2024

Extensions

a(7)-a(14) from Andrew Howroyd, Apr 14 2016

A068392 Number of partitions of n X n checkerboard into two edgewise-connected sets, counting partitions equal under rotation or reflection only once.

Original entry on oeis.org

0, 2, 11, 92, 2100, 140834
Offset: 1

Views

Author

R. H. Hardin, Mar 03 2002

Keywords

Crossrefs

Cf. A068381, A068393.

A068393 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. Partitions equal under rotation or reflection are counted only once.

Original entry on oeis.org

2, 3, 7, 44, 494, 748827, 99987552, 23904291912, 23904291912, 14647978829979, 16186345621426754, 45843626565163628751, 235646717730827228414584, 3099290829556018890177304005
Offset: 2

Views

Author

R. H. Hardin, Mar 03 2002

Keywords

Comments

For even n > 2 the only symmetry possible is rotation by 180 degrees. For odd n > 1 the only symmetries are reflections either horizontally or vertically. - Andrew Howroyd, Apr 15 2016

Examples

			From _Andrew Howroyd_, Apr 15 2016: (Start)
Case n=4: There are 2 nonisomorphic symmetrical solutions (see illustration below). a(4)=(A068381(4)/8 + 2)/2 = 7.
    __.__.__.__.    __.__.__.__.
   |   __    __|   |   __   |  |
   |  |  |  |  |   |  |  |  |  |
   |__|  |__|  |   |  |  |__|  |
   |__.__.__.__|   |__|__.__.__|
Case n=5: There are 7 nonisomorphic symmetrical solutions (see illustration below). a(5)=(A068381(5)/8 + 7)/2 = 44.
    __.__.__.__.__.   __.__.__.__.__.   __.__.__.__.__.   __.__.__.__.__.
   |   __|  |__   |  |   __|  |__   |  |  |__    __|  |  |  |   __   |  |
   |  |__    __|  |  |  |   __   |  |  |   __|  |__   |  |  |  |  |  |  |
   |   __|  |__   |  |  |  |  |  |  |  |  |   __   |  |  |  |  |  |  |  |
   |  |__.__.__|  |  |  |__|  |__|  |  |  |__|  |__|  |  |  |__|  |__|  |
   |__.__.__.__.__|  |__.__.__.__.__|  |__.__.__.__.__|  |__.__.__.__.__|
    __.__.__.__.__.   __.__.__.__.__.   __.__.__.__.__.
   |__.__    __.__|  |__    __    __|  |   __    __   |
   |   __|  |__   |  |  |  |  |  |  |  |__|  |  |  |__|
   |  |   __   |  |  |  |  |  |  |  |  |   __|  |__   |
   |  |__|  |__|  |  |  |__|  |__|  |  |  |__.__.__|  |
   |__.__.__.__.__|  |__.__.__.__.__|  |__.__.__.__.__|
(End)

Crossrefs

Cf. A068381, A068416, A068392, A265914.

Extensions

a(7)-a(15) from Andrew Howroyd, Apr 15 2016
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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