A212209 -id:A212209 - cp's OEIS frontend

A068305 1/16 the number of colorings of an n X n octagonal array with 16 colors.

1, 2730, 1097599230, 65013773510046270, 567344965666217922692546310, 729405912578031916581095228654013174510, 138156586653036397048665899068285784224754055444205830, 3855264256604335001270711977271948484283006596889356139648293174199350

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Cf. A068239-A068305, A000332, A002417, A027441, A212208, A212209.

a(5)-a(8) from Alois P. Heinz, May 04 2012

A212208 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the square diagonal grid graph DG_(n,n), highest powers first.

Original entry on oeis.org

1, 0, 1, -6, 11, -6, 0, 1, -20, 174, -859, 2627, -5082, 6048, -4023, 1134, 0, 1, -42, 825, -10054, 85011, -528254, 2491825, -9084089, 25795983, -57031153, 97292827, -125639547, 118705077, -77301243, 30931875, -5709042, 0, 1, -72, 2492, -55183, 877812

Offset: 1

Alois P. Heinz, May 04 2012

The square diagonal grid graph DG_(n,n) has n^2 = A000290(n) vertices and 2*(n-1)*(2*n-1) = A002943(n-1) edges. The chromatic polynomial of DG_(n,n) has n^2+1 = A002522(n) coefficients.

			3 example graphs:                          o---o---o
.                                          |\ /|\ /|
.                                          | X | X |
.                                          |/ \|/ \|
.                             o---o        o---o---o
.                             |\ /|        |\ /|\ /|
.                             | X |        | X | X |
.                             |/ \|        |/ \|/ \|
.                o            o---o        o---o---o
Graph:        DG_(1,1)       DG_(2,2)       DG_(3,3)
Vertices:        1              4              9
Edges:           0              6             20
The square diagonal grid graph DG_(2,2) equals the complete graph K_4 and has chromatic polynomial q*(q-1)*(q-2)*(q-3) = q^4 -6*q^3 +11*q^2 -6*q => row 2 = [1, -6, 11, -6, 0].
Triangle T(n,k) begins:
1,    0;
1,   -6,    11,      -6,        0;
1,  -20,   174,    -859,     2627,      -5082, ...
1,  -42,   825,  -10054,    85011,    -528254, ...
1,  -72,  2492,  -55183,   877812,  -10676360, ...
1, -110,  5895, -205054,  5203946, -102687204, ...
1, -156, 11946, -598491, 22059705, -637802510, ...

Alois P. Heinz, Rows n = 1..7, flattened
Wikipedia, Chromatic Polynomial

Columns 1-2 give: A000012, (-1)*A002943(n-1).
Row sums (for n>1) and last elements of rows give: A000004, row lengths give: A002522.
Cf. A000290, A212209.

A208021 T(n,k) = Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal, vertical, diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 7, 7, 5, 15, 87, 270, 87, 15, 52, 1657, 27093, 27093, 1657, 52, 203, 43833, 5252041, 30066912, 5252041, 43833, 203, 877, 1515903, 1688298227, 80318704605, 80318704605, 1688298227, 1515903, 877, 4140, 65766991

Offset: 1

R. H. Hardin, Feb 22 2012

Equivalently, the number of colorings of the n x k king graph using any number of colors up to permutation of the colors. - Andrew Howroyd, Jun 25 2017

			Table starts
...1........1............2...............5...............15..............52
...1........1............7..............87.............1657...........43833
...2........7..........270...........27093..........5252041......1688298227
...5.......87........27093........30066912......80318704605.421673189900658
..15.....1657......5252041.....80318704605.3662498214110836
..52....43833...1688298227.421673189900658
.203..1515903.819147302097
.877.65766991
...
Some solutions for n=4 k=3
..0..1..2....0..1..0....0..1..0....0..1..0....0..1..0....0..1..0....0..1..2
..2..3..4....2..3..2....2..3..2....2..3..2....2..3..2....2..3..2....2..3..0
..5..6..0....4..0..4....0..1..0....4..1..0....0..1..0....0..4..0....4..5..4
..2..3..1....1..2..1....2..3..4....5..2..3....2..4..2....1..2..1....0..1..2

Andrew Howroyd, Table of n, a(n) for n = 1..231 (terms 1..49 from R. H. Hardin)
Eric Weisstein's World of Mathematics, King Graph

Columns 1-5 are A000110(n-1), A020556(n-1), A208018, A208019, A208020.
Main diagonal is A289136.
Cf. A207868, A212208, A212209.

A289136 Number of colorings of the n X n king graph up to permutation of the colors.

Original entry on oeis.org

1, 1, 270, 30066912, 3662498214110836, 978788002444637886853083440, 1017523795194980592656592864724960780556190, 6723457445689415320074916040888682277646129463792942126176174

Offset: 1

Andrew Howroyd, Jun 25 2017

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..11
Eric Weisstein's World of Mathematics, King Graph

Main diagonal of A208021.

Cf. A212208, A212209.

A208021 = Import["https://oeis.org/A208021/b208021.txt", "Table"][[All, 2]];
a[n_] := A208021[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 11}] (* Jean-François Alcover, Sep 23 2019 *)

A068250 1/24 the number of colorings of a 3 X 3 octagonal array with n colors.

Original entry on oeis.org

3, 280, 5265, 47040, 271250, 1170288, 4105710, 12334080, 32837805, 79365000, 177200023, 370319040, 731732820, 1377981920, 2488927500, 4334174208, 7307669895, 11972250360, 19116135885, 29823640000, 45562619718, 68291480400, 100588847450, 145809331200

Offset: 4

R. H. Hardin, Feb 24 2002

Alois P. Heinz, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A068239-A068305, A000332, A002417, A027441, A212208, A212209.

[n*(n-3)^4*(n^4-8*n^3+24*n^2-31*n+14)/24: n in [4..27]]; // Bruno Berselli, May 04 2012

a:= n-> (n-2)*(n-1)*(n^2-5*n+7)*(n-3)^4*n/24:
seq(a(n), n=4..40);  # Alois P. Heinz, May 04 2012

From Alois P. Heinz, May 04 2012: (Start)
G.f.: (832*x^5+4805*x^4+6630*x^3+2600*x^2+250*x+3)*x^4 / (x-1)^10.
a(n) = (n^9 -20*n^8 +174*n^7 -859*n^6 +2627*n^5 -5082*n^4 +6048*n^3 -4023*n^2 +1134*n)/24. (End)

A068252 1/24 the number of colorings of a 5 X 5 octagonal array with n colors.

Original entry on oeis.org

15, 13544240, 138446293905, 122914516488960, 25999554922206250, 2153128175383358880, 92188903926678792030, 2420834781123276840960, 43619102772258066624945, 582825001391348708850000, 6107144216640392108077895, 52317129545418639168933120

Offset: 4

R. H. Hardin, Feb 24 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..1000
Alois P. Heinz, Formula for A068252

Cf. A068239-A068305, A000332, A002417, A027441, A212208, A212209.

a:= n-> (749775400182+ (-5612178463551+ (20061456442182+ (-45672704671212+ (74464103614803+ (-92636079268387+ (91447345270476+ (-73522860568402+ (49016457219873+ (-27440963053418+ (13013052032729+ (-5257328793792+
(1815373485182+ (-536334824998+ (135430280705+ (-29134045002+ (5309217327+ (-812594023+ (103170636+ (-10676360+ (877812+ (-55183+ (2492+(-72+n)*n) *n)*n) *n)*n) *n)*n) *n)*n) *n)*n) *n)*n) *n)*n) *n) *n) *n) *n) *n) *n) *n) *n) *n/24:
seq(a(n), n=4..20); #  Alois P. Heinz, May 04 2012

Extended beyond a(9) by Alois P. Heinz, May 04 2012

A068294 1/5 the number of colorings of an n X n octagonal array with 5 colors.

Original entry on oeis.org

1, 24, 1344, 187008, 65012352, 56638464768, 123968096842368, 682463815024774272, 9455897825697924931968, 329835949030491781985964672, 28967729014957963606903092579840, 6405563550899359667890368716213003520, 3566277295414164369700428069692324488565760, 4998812913455162820842387241261713638306330756608

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Christian Gaetz and Yibo Gao, Interlacing triangles, Schubert puzzles, and graph colorings, arXiv:2408.07863 [math.CO], 2024. See p. 15. See also Sém. Lothar. Combin., (2025) Vol. 93B, Art. No. 26. See p. 10.

Cf. A068239-A068305, A000332, A002417, A027441, A212208, A212209.

a(8) from Alois P. Heinz, May 03 2012

a(9)-a(14) from Andrew Howroyd, Jun 25 2017

A068295 1/6 the number of colorings of an n X n octagonal array with 6 colors.

Original entry on oeis.org

1, 60, 21060, 44172540, 553785175620, 41503847098683420, 18595665812583520041060, 49808667823922052573513153300, 797553247913411612275102392386088300, 76343133584713932892876357250654965547341140, 43684986787209742051887297631700585399912369700067860

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Cf. A068239-A068305, A000332, A002417, A027441, A212208, A212209.

a(7)-a(8) from Alois P. Heinz, May 04 2012

a(9)-a(11) from Andrew Howroyd, Jun 25 2017

A068296 1/7 the number of colorings of an n X n octagonal array with 7 colors.

Original entry on oeis.org

1, 120, 161280, 2451486720, 421421199390720, 819296954885530030080, 18013654371239090908333148160, 4479149596637688119857975807616194560, 12595676735919821000208123310178819903753287680, 400570738126155220679966449710339235598075113573822617600

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Cf. A068239-A068305, A000332, A002417, A027441, A212208, A212209.

a(6)-a(8) from Alois P. Heinz, May 04 2012

a(9)-a(10) from Andrew Howroyd, Jun 25 2017

A068297 1/8 the number of colorings of an n X n octagonal array with 8 colors.

Original entry on oeis.org

1, 210, 813750, 58494843750, 77998664766618750, 1929292299321647690193750, 885215793907124317300191394818750, 7534251364260636630211044759506294857668750

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Cf. A068239-A068305, A000332, A002417, A027441, A212208, A212209.

a(6)-a(8) from Alois P. Heinz, May 04 2012

cp's OEIS Frontend

A068305 1/16 the number of colorings of an n X n octagonal array with 16 colors.

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A212208 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the square diagonal grid graph DG_(n,n), highest powers first.

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A208021 T(n,k) = Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal, vertical, diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

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A289136 Number of colorings of the n X n king graph up to permutation of the colors.

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Mathematica

A068250 1/24 the number of colorings of a 3 X 3 octagonal array with n colors.

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Magma

Maple

Formula

A068252 1/24 the number of colorings of a 5 X 5 octagonal array with n colors.

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Maple

Extensions

A068294 1/5 the number of colorings of an n X n octagonal array with 5 colors.

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A068295 1/6 the number of colorings of an n X n octagonal array with 6 colors.

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A068296 1/7 the number of colorings of an n X n octagonal array with 7 colors.

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Extensions

A068297 1/8 the number of colorings of an n X n octagonal array with 8 colors.

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Author

Keywords

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Extensions