A216084 -id:A216084 - cp's OEIS frontend

A216078 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X 2 array with new integer colors introduced in row major order.

1, 1, 3, 7, 27, 87, 409, 1657, 9089, 43833, 272947, 1515903, 10515147, 65766991, 501178937, 3473600465, 28773452321, 218310229201, 1949230218691, 16035686850327, 153281759047387, 1356791248984295, 13806215066685433, 130660110400259849, 1408621900803060705

Offset: 1

R. H. Hardin, Sep 01 2012

nonn

Number of vertex covers and independent vertex sets of the n-1 X n-1 white bishops graph. Equivalently, the number of ways to place any number of non-attacking bishops on the white squares of an n-1 X n-1 board. - Andrew Howroyd, May 08 2017

Number of pairs of partitions (A<=B) of [n-1] such that the nontrivial blocks of A are of type {k,n-1-k} if n is even, and of type {k,n-k} if n is odd. - Francesca Aicardi, May 28 2022

			Some solutions for n = 5:
  x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0
  1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x
  x 2   x 0   x 0   x 2   x 0   x 1   x 1   x 2   x 2   x 1
  0 x   2 x   1 x   3 x   1 x   0 x   2 x   3 x   0 x   0 x
  x 3   x 1   x 2   x 2   x 0   x 1   x 1   x 1   x 2   x 0
There are 4 white squares on a 3 X 3 board. There is 1 way to place no non-attacking bishops, 4 ways to place 1 and 2 ways to place 2 so a(4) = 1 + 4 + 2 = 7. - _Andrew Howroyd_, Jun 06 2017

R. H. Hardin, Table of n, a(n) for n = 1..210
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Eric Weisstein's World of Mathematics, White Bishop Graph

Column 2 of A216084.
Row sums of A274106(n-1).
Cf. A020556, A094577, A216332, A201862, A286423.

a:= n-> (m-> add(binomial(m, k)*combinat[bell](m+k+e)
           , k=0..m))(iquo(n-1, 2, 'e')):
seq(a(n), n=1..26);  # Alois P. Heinz, Oct 03 2022

a[n_] := Module[{m, e}, {m, e} = QuotientRemainder[n - 1, 2];
   Sum[Binomial[m, k]*BellB[m + k + e], {k, 0, m}]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jul 25 2022, after Francesca Aicardi *)

a(n) = Sum_{k=0..m} binomial(m, k)*Bell(m+k+e), with m = floor((n-1)/2), e = (n+1) mod 2 and where Bell(n) is the Bell exponential number A000110(n). - Francesca Aicardi, May 28 2022
From Vaclav Kotesovec, Jul 29 2022: (Start)
a(2*k) = A020556(k).
a(2*k+1) = A094577(k). (End)

A216079 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X 3 array with new integer colors introduced in row major order.

Original entry on oeis.org

1, 2, 5, 52, 203, 4140, 21147, 678570, 4213597, 190899322, 1382958545, 82864869804, 682076806159, 51724158235372, 474869816156751, 44152005855084346, 445958869294805289, 49631246523618756274, 545717047936059989389

Offset: 1

R. H. Hardin, Sep 01 2012

nonn

Column 3 of A216084.

Appears to be identical to A000110 except for elements in the latter indexed by 3x + 2 for x >= 0. - Chris Boyd, Dec 20 2015

			Some solutions for n=4:
..x..0..x....x..0..x....x..0..x....x..0..x....x..0..x....x..0..x....x..0..x
..1..x..0....1..x..2....1..x..2....1..x..2....1..x..0....1..x..0....1..x..0
..x..2..x....x..3..x....x..0..x....x..0..x....x..1..x....x..1..x....x..2..x
..3..x..2....4..x..3....3..x..0....2..x..0....2..x..3....2..x..1....3..x..0

R. H. Hardin, Table of n, a(n) for n = 1..210

A216080 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an nX4 array with new integer colors introduced in row major order.

Original entry on oeis.org

1, 4, 42, 602, 15144, 433252, 18378580, 839621092, 52183667212, 3404551437592, 286529186194664, 24947253698767632, 2701657928740913416, 299900819552115769424, 40316833236311531705440, 5520994078882300886182032

Offset: 1

R. H. Hardin Sep 01 2012

nonn

Column 4 of A216084

			Some solutions for n=4
..x..0..x..1....x..0..x..1....x..0..x..1....x..0..x..1....x..0..x..1
..2..x..3..x....2..x..3..x....2..x..3..x....1..x..2..x....2..x..3..x
..x..1..x..2....x..4..x..5....x..4..x..1....x..3..x..1....x..1..x..0
..3..x..0..x....6..x..0..x....5..x..3..x....0..x..4..x....4..x..3..x

R. H. Hardin, Table of n, a(n) for n = 1..78

A216081 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X 5 array with new integer colors introduced in row major order.

Original entry on oeis.org

1, 12, 136, 12840, 373780, 99827160, 5116799368, 2751754706832, 212339816996032, 194478305002268064, 20707051422399128416, 29191372156840204219200, 4055642578974004982217280, 8220961603626446873042001408

Offset: 1

R. H. Hardin, Sep 01 2012

nonn

Column 5 of A216084.

			Some solutions for n=4
..x..0..x..1..x....x..0..x..1..x....x..0..x..1..x....x..0..x..1..x
..2..x..3..x..4....2..x..3..x..4....1..x..0..x..2....1..x..2..x..1
..x..5..x..2..x....x..5..x..3..x....x..2..x..3..x....x..3..x..4..x
..1..x..6..x..1....6..x..5..x..1....1..x..4..x..1....4..x..5..x..0

R. H. Hardin, Table of n, a(n) for n = 1..49

Cf. A216084.

A216082 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an nX6 array with new integer colors introduced in row major order.

Original entry on oeis.org

2, 34, 2440, 322768, 87852160, 33093252880, 20358402201792, 15604440074084672, 17423570614012565376, 22946861101272125055616, 41082882676098706722895360, 83889915459846520832774898688, 222247072052822589353800809064448

Offset: 1

R. H. Hardin Sep 01 2012

nonn

Column 6 of A216084

			Some solutions for n=4
..x..0..x..1..x..0....x..0..x..1..x..2....x..0..x..1..x..2....x..0..x..1..x..2
..1..x..2..x..3..x....1..x..3..x..0..x....2..x..3..x..0..x....1..x..3..x..4..x
..x..3..x..4..x..5....x..1..x..4..x..1....x..0..x..4..x..5....x..5..x..3..x..6
..1..x..3..x..1..x....4..x..3..x..5..x....3..x..0..x..6..x....3..x..4..x..3..x

R. H. Hardin, Table of n, a(n) for n = 1..26

A216083 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an nX7 array with new integer colors introduced in row major order.

Original entry on oeis.org

2, 136, 10900, 12297870, 4044766240, 18383343930862, 14216413305846436, 166176901557427161788, 238921763468063076499860, 5737167226254078112766902750, 13446078759879900390396507182880

Offset: 1

R. H. Hardin Sep 01 2012

nonn

Column 7 of A216084

			Some solutions for n=4
..x..0..x..1..x..2..x....x..0..x..1..x..0..x....x..0..x..1..x..0..x
..1..x..0..x..3..x..4....1..x..0..x..2..x..1....1..x..0..x..2..x..3
..x..4..x..0..x..5..x....x..3..x..0..x..4..x....x..2..x..0..x..4..x
..0..x..2..x..1..x..2....0..x..2..x..3..x..4....3..x..4..x..3..x..1

R. H. Hardin, Table of n, a(n) for n = 1..17

A216085 Number of horizontal and antidiagonal neighbor colorings of the odd squares of a 3Xn array with new integer colors introduced in row major order.

Original entry on oeis.org

1, 3, 5, 42, 136, 2440, 10900, 322776, 1805632, 77360796, 514264344, 29562890886, 226109935520, 16604827619828, 142953007039492, 12954736337254696, 123560080110698296, 13465808369740333664, 140585883811112551960

Offset: 1

R. H. Hardin Sep 01 2012

nonn

Row 3 of A216084

			Some solutions for n=4
..x..0..x..1....x..0..x..1....x..0..x..1....x..0..x..1....x..0..x..1
..2..x..0..x....1..x..0..x....2..x..0..x....2..x..3..x....2..x..3..x
..x..1..x..2....x..2..x..0....x..2..x..1....x..0..x..3....x..2..x..3

R. H. Hardin, Table of n, a(n) for n = 1..41

A216086 Number of horizontal and antidiagonal neighbor colorings of the odd squares of a 4Xn array with new integer colors introduced in row major order.

Original entry on oeis.org

2, 7, 52, 602, 12840, 322768, 12297870, 514264344, 29562890980, 1804957901332, 142953068558998, 11807624690799832, 1216988850964527370, 129339015912912689012, 16679506383419809650408, 2201178485290778203106968

Offset: 1

R. H. Hardin Sep 01 2012

nonn

Row 4 of A216084

			Some solutions for n=4
..x..0..x..1....x..0..x..1....x..0..x..1....x..0..x..1....x..0..x..1
..1..x..2..x....1..x..2..x....2..x..3..x....2..x..0..x....2..x..0..x
..x..0..x..2....x..3..x..4....x..0..x..4....x..3..x..4....x..1..x..2
..1..x..0..x....5..x..6..x....2..x..0..x....5..x..6..x....3..x..0..x

R. H. Hardin, Table of n, a(n) for n = 1..19

A216077 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X n array with new integer colors introduced in row major order.

Original entry on oeis.org

1, 1, 5, 602, 373780, 33093252880, 14216413305846436, 2562708724272040433794232, 2209363619821632289359062591728192

Offset: 1

R. H. Hardin Sep 01 2012

nonn

Diagonal of A216084

			Some solutions for n=4
..x..0..x..1....x..0..x..1....x..0..x..1....x..0..x..1....x..0..x..1
..2..x..3..x....2..x..3..x....2..x..3..x....2..x..0..x....2..x..3..x
..x..4..x..3....x..2..x..4....x..2..x..3....x..3..x..4....x..1..x..3
..3..x..4..x....0..x..2..x....0..x..2..x....5..x..3..x....3..x..4..x

A216087 Number of horizontal and antidiagonal neighbor colorings of the odd squares of a 5Xn array with new integer colors introduced in row major order.

Original entry on oeis.org

2, 27, 203, 15144, 373780, 87852160, 4044766240, 2004646634218, 142953007126612, 123560104158582188, 12361734369562410736

Offset: 1

R. H. Hardin Sep 01 2012

nonn

Row 5 of A216084

			Some solutions for n=4
..x..0..x..1....x..0..x..1....x..0..x..1....x..0..x..1....x..0..x..1
..2..x..3..x....2..x..3..x....2..x..3..x....2..x..3..x....2..x..3..x
..x..1..x..2....x..2..x..3....x..0..x..4....x..4..x..0....x..4..x..5
..3..x..0..x....0..x..4..x....1..x..3..x....1..x..5..x....0..x..4..x
..x..1..x..3....x..2..x..1....x..5..x..4....x..1..x..2....x..6..x..3

cp's OEIS Frontend

A216078 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X 2 array with new integer colors introduced in row major order.

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A216079 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X 3 array with new integer colors introduced in row major order.

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A216080 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an nX4 array with new integer colors introduced in row major order.

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A216081 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X 5 array with new integer colors introduced in row major order.

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A216082 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an nX6 array with new integer colors introduced in row major order.

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A216083 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an nX7 array with new integer colors introduced in row major order.

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A216085 Number of horizontal and antidiagonal neighbor colorings of the odd squares of a 3Xn array with new integer colors introduced in row major order.

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A216086 Number of horizontal and antidiagonal neighbor colorings of the odd squares of a 4Xn array with new integer colors introduced in row major order.

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A216077 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X n array with new integer colors introduced in row major order.

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A216087 Number of horizontal and antidiagonal neighbor colorings of the odd squares of a 5Xn array with new integer colors introduced in row major order.

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