A212195 -id:A212195 - cp's OEIS frontend

A212163 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the rhombic hexagonal square grid graph RH_(k,k).

1, 0, 2, 0, 0, 3, 0, 0, 6, 4, 0, 0, 6, 48, 5, 0, 0, 6, 1056, 180, 6, 0, 0, 6, 45696, 32940, 480, 7, 0, 0, 6, 4034304, 30847500, 393600, 1050, 8, 0, 0, 6, 739642368, 148039757460, 3312560640, 2735250, 2016, 9

Offset: 1

Alois P. Heinz, May 02 2012

The rhombic hexagonal square grid graph RH_(n,n) has n^2 = A000290(n) vertices and (n-1)*(3*n-1) = A045944(n-1) edges; see A212162 for example. The chromatic polynomial of RH_(n,n) has n^2+1 = A002522(n) coefficients.

A differs from A212195 first at (n,k) = (4,5): A(4,5) = 4034304, A212195(4,5) = 4038432.

			Square array A(n,k) begins:
  1,    0,       0,            0,                 0, ...
  2,    0,       0,            0,                 0, ...
  3,    6,       6,            6,                 6, ...
  4,   48,    1056,        45696,           4034304, ...
  5,  180,   32940,     30847500,      148039757460, ...
  6,  480,  393600,   3312560640,   286169360240640, ...
  7, 1050, 2735250, 123791435250, 97337270132408250, ...

Andrew Howroyd, Table of n, a(n) for n = 1..153
Wikipedia, Chromatic Polynomial

Columns k=1-6 give: A000027, A047927(n) = 6*A002417(n-2), 6*A068244, 6*A068245, 6*A068246, 6*A068247.
Rows n=1-15 give: A000007, A000038, A040006, 4*A068271, 5*A068272, 6*A068273, 7*A068274, 8*A068275, 9*A068276, 10*A068277, 11*A068278, 12*A068279, 13*A068280, 14*A068281, 15*A068282.
Cf. A212162, A212195, A208054, A208050.

A068248 1/6 the number of colorings of a 5 X 5 staggered hexagonal array with n colors.

Original entry on oeis.org

1, 673072, 24674450670, 47695073906240, 16222886703881375, 1842996310592836896, 98798502888215704812, 3068393794369671728640, 62960689505171989129005, 933100312771109288146000, 10639781342848431789710266, 97779035987698387480546752, 750090455960001686602653035

Offset: 3

R. H. Hardin, Feb 24 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Alois P. Heinz, Formula for A068248

Cf. A068239-A068305, A000332, A002417, A027441, A212194, A212195.

a:= n-> (3006792824+ (-26845691044+ (115537440058+ (-319333174471+ (636781496832+ (-975359012827+ (1192518013138+ (-1193724499144+ (995462037353+ (-699932345254+ (418375639535+ (-213720396671+ (93568819963+ (-35133625647+ (11298632584+
(-3101089710+ (722137763+ (-141421592+ (23000726+ (-3051871+ (321994+ (-25992+ (1508+(-56+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/6:
seq (a(n), n=3..40);  # Alois P. Heinz, May 03 2012

Extended beyond a(10) by Alois P. Heinz, May 03 2012

A212194 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the staggered hexagonal square grid graph SH_(n,n), highest powers first.

Original entry on oeis.org

1, 0, 1, -5, 8, -4, 0, 1, -16, 112, -448, 1120, -1791, 1786, -1012, 248, 0, 1, -33, 510, -4898, 32703, -160859, 602408, -1749715, 3975561, -7068408, 9755858, -10265148, 7968348, -4304712, 1445104, -226720, 0, 1, -56, 1508, -25992, 321994, -3051871, 23000726, -141421592, 722137763, -3101089710

Offset: 1

Alois P. Heinz, May 03 2012

T differs from A212162 first at (n,k) = (5,10): T(5,10) = -3101089710, A212162(5,10) = -3101089711.

The staggered hexagonal square grid graph SH_(n,n) has n^2 = A000290(n) vertices and (n-1)*(3*n-1) = A045944(n-1) edges. The chromatic polynomial of SH_(n,n) has n^2+1 = A002522(n) coefficients.

			3 example graphs:                        o--o--o
.                                        | /|\ |
.                                        |/ | \|
.                            o--o        o--o--o
.                            | /|        | /|\ |
.                            |/ |        |/ | \|
.               o            o--o        o--o--o
Graph:       SH_(1,1)      SH_(2,2)      SH_(3,3)
Vertices:       1             4             9
Edges:          0             5            16
The staggered hexagonal square grid graph SH_(2,2) has chromatic polynomial q^4 -5*q^3 +8*q^2 -4*q => row 2 = [1, -5, 8, -4, 0].
Triangle T(n,k) begins:
1,    0;
1,   -5,     8,      -4,        0;
1,  -16,   112,    -448,     1120,      -1791, ...
1,  -33,   510,   -4898,    32703,    -160859, ...
1,  -56,  1508,  -25992,   321994,   -3051871, ... , -3101089710, ...
1,  -85,  3520,  -94620,  1855860,  -28306676, ...
1, -120,  7068, -272344,  7720110, -171656543, ...
1, -161, 12782, -667058, 25738055, -783003395, ...

Alois P. Heinz, Rows n = 1..8, flattened
Wikipedia, Chromatic Polynomial

Columns 1-2 give: A000012, (-1)*A045944(n-1).
Row sums (for n>1) and last elements of rows give: A000004, row lengths give: A002522.
Cf. A000290, A212162, A212195.

A068249 1/6 the number of colorings of a 6 X 6 staggered hexagonal array with n colors.

Original entry on oeis.org

1, 123933504, 607054532700510, 42295002762341131520, 221621606435023540827375, 227499908955471802467974016, 76749558468512064584917217484, 11559255724071428199019433809920, 945121792213234612017996840421965, 47886238153764555711259823337784000

Offset: 3

R. H. Hardin, Feb 24 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Alois P. Heinz, Formula for A068249

Cf. A068239-A068305, A000332, A002417, A027441, A212194, A212195.

Extended beyond a(7) by Alois P. Heinz, May 03 2012

A068283 1/4 the number of colorings of an n X n staggered hexagonal array with 4 colors.

Original entry on oeis.org

1, 12, 264, 11424, 1009608, 185900256, 71883991272, 58646051219616, 101273697848025960, 370689919380760915680, 2880341312871966116331528, 47562687666338563715921774496

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Terms for rhombic- and staggered- hexagonal arrays are the same for n in 1..4.

Cf. A068239-A068305, A000332, A002417, A027441, A212194, A212195.

a(9) from Alois P. Heinz, May 03 2012

a(10)-a(12) from Sean A. Irvine, Feb 07 2024

A068284 1/5 the number of colorings of an n X n staggered hexagonal array with 5 colors.

Original entry on oeis.org

1, 36, 6588, 6169500, 29609340804, 728465439240612, 91873868538291559236, 59398217444060775978706980, 196856852524443536659406407297548

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Terms for rhombic- and staggered- hexagonal arrays are the same for n in 1..4.

Cf. A068239-A068305, A000332, A002417, A027441, A212194, A212195.

a(8)-a(9) from Alois P. Heinz, May 03 2012

A068285 1/6 the number of colorings of an n X n staggered hexagonal array with 6 colors.

Original entry on oeis.org

1, 80, 65600, 552093440, 47695073906240, 42295002762341131520, 384997602083152626170331200, 35973286295593063089098993201626880, 34502872879799250159432202960093141227521600

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Terms for rhombic- and staggered- hexagonal arrays are the same for n in 1..4.

Cf. A068239-A068305, A000332, A002417, A027441, A212194, A212195.

a(7)-a(9) from Alois P. Heinz, May 03 2012

A068286 1/7 the number of colorings of an n X n staggered hexagonal array with 7 colors.

Original entry on oeis.org

1, 150, 390750, 17684490750, 13905331460469750, 189961376944305892137750, 45086270412704540409445089549750, 185916670055945278226619825124072763091750, 13319483009963134380534870853035913218901916338644750

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Terms for rhombic- and staggered- hexagonal arrays are the same for n in 1..4.

Cf. A068239-A068305, A000332, A002417, A027441, A212194, A212195.

a(7)-a(9) from Alois P. Heinz, May 03 2012

A068287 1/8 the number of colorings of an n X n staggered hexagonal array with 8 colors.

Original entry on oeis.org

1, 252, 1679832, 296251234272, 1382247232944627672, 170624931716603851850980512, 557224793178655263880694805929352312, 48144818121088035723665058177874441292104576992, 110052483236400191824645806778218519319800722550874233268152

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Terms for rhombic- and staggered- hexagonal arrays are the same for n in 1..4.

Cf. A068239-A068305, A000332, A002417, A027441, A212194, A212195.

a(6)-a(9) from Alois P. Heinz, May 03 2012

A068288 1/9 the number of colorings of an n X n staggered hexagonal array with 9 colors.

Original entry on oeis.org

1, 392, 5765144, 3181273031384, 65865668592143803208, 51166372312341376389944811656, 1491342882335672198457853495232503998152, 1630939089590623060636571472516037785233254346536136, 66921448813902442793929754551432219305219935812987218443538247160

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Terms for rhombic- and staggered- hexagonal arrays are the same for n in 1..4.

Cf. A068239-A068305, A000332, A002417, A027441, A212194, A212195.

a(6)-a(9) from Alois P. Heinz, May 03 2012

cp's OEIS Frontend

A212163 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the rhombic hexagonal square grid graph RH_(k,k).

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A068248 1/6 the number of colorings of a 5 X 5 staggered hexagonal array with n colors.

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A212194 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the staggered hexagonal square grid graph SH_(n,n), highest powers first.

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A068249 1/6 the number of colorings of a 6 X 6 staggered hexagonal array with n colors.

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A068283 1/4 the number of colorings of an n X n staggered hexagonal array with 4 colors.

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A068284 1/5 the number of colorings of an n X n staggered hexagonal array with 5 colors.

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A068285 1/6 the number of colorings of an n X n staggered hexagonal array with 6 colors.

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A068286 1/7 the number of colorings of an n X n staggered hexagonal array with 7 colors.

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A068287 1/8 the number of colorings of an n X n staggered hexagonal array with 8 colors.

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A068288 1/9 the number of colorings of an n X n staggered hexagonal array with 9 colors.

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