A212163 -id:A212163 - cp's OEIS frontend

A212162 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the rhombic hexagonal square grid graph RH_(n,n), highest powers first.

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, -3101089711

Offset: 1

Alois P. Heinz, May 02 2012

T differs from A212194 first at (n,k) = (5,10): T(5,10) = -3101089711, A212194(5,10) = -3101089710.

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. The chromatic polynomial of RH_(n,n) has n^2+1 = A002522(n) coefficients.

			3 example graphs:                        o--o--o
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.               o            o--o        o--o--o
Graph:       RH_(1,1)      RH_(2,2)      RH_(3,3)
Vertices:       1             4             9
Edges:          0             5            16
The rhombic hexagonal square grid graph RH_(2,2) has chromatic polynomial q*(q-1)*(q-2)^2 = 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, ... , -3101089711, ...
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, A212163, A212194.

A212195 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the staggered hexagonal square grid graph SH_(k,k).

Original entry on oeis.org

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, 4038432, 30847500, 393600, 1050, 8, 0, 0, 6, 743601024, 148046704020, 3312560640, 2735250, 2016, 9

Offset: 1

Alois P. Heinz, May 03 2012

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; see A212194 for example. The chromatic polynomial of SH_(n,n) has n^2+1 = A002522(n) coefficients.

A differs from A212163 first at (n,k) = (4,5): A(4,5) = 4038432, A212163(4,5) = 4034304.

			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,           4038432, ...
  5,  180,   32940,     30847500,      148046704020, ...
  6,  480,  393600,   3312560640,   286170443437440, ...
  7, 1050, 2735250, 123791435250, 97337320223288250, ...

Wikipedia, Chromatic Polynomial

Columns k=1-6 give: A000027, A047927(n) = 6*A002417(n-2), 6*A068244, 6*A068245, 6*A068248, 6*A068249.
Rows n=1-10, 16-18 give: A000007, A000038, A040006, 4*A068283, 5*A068284, 6*A068285, 7*A068286, 8*A068287, 9*A068288, 10*A068289, 16*A068290, 17*A068291, 18*A068292.
Cf. A212163, A212194.

A208050 T(n,k)=Number of nXk 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 8, 8, 5, 14, 32, 44, 32, 14, 41, 128, 244, 244, 128, 41, 122, 512, 1356, 1904, 1356, 512, 122, 365, 2048, 7540, 14976, 14976, 7540, 2048, 365, 1094, 8192, 41932, 118096, 168096, 118096, 41932, 8192, 1094, 3281, 32768, 233204, 931968

Offset: 1

R. H. Hardin, Feb 22 2012

Equivalently, the number of colorings in the rhombic hexagonal square grid graph RH_(n,k) using 4 colors up to permutation of the colors. - Andrew Howroyd, Jun 25 2017

			Table starts
...1....1......2.......5........14.........41..........122...........365
...1....2......8......32.......128........512.........2048..........8192
...2....8.....44.....244......1356.......7540........41932........233204
...5...32....244....1904.....14976.....118096.......931968.......7356288
..14..128...1356...14976....168096....1897888.....21472544.....243113056
..41..512...7540..118096...1897888...30818432....502504448....8206614784
.122.2048..41932..931968..21472544..502504448..11838995200..279733684992
.365.8192.233204.7356288.243113056.8206614784.279733684992.9578237457408
...
Some solutions for n=4 k=3
..0..1..0....0..1..2....0..1..0....0..1..0....0..1..2....0..1..2....0..1..2
..2..3..1....2..3..0....2..3..1....2..3..1....2..0..3....2..3..0....2..0..3
..1..2..0....0..1..2....0..2..3....0..2..3....1..2..1....1..2..1....1..2..0
..3..1..2....2..3..1....1..0..1....3..0..2....3..0..3....3..0..2....3..1..2

R. H. Hardin, Table of n, a(n) for n = 1..312

Columns 1-7 are A007051(n-2), A004171(n-2), A208044, A208046, A208047-A208049.
Main diagonal is A208045.
Cf. A208054, A068271, A212162, A212163.

A208054 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 or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 15, 15, 5, 15, 203, 716, 203, 15, 52, 4140, 83440, 83440, 4140, 52, 203, 115975, 18171918, 112073062, 18171918, 115975, 203, 877, 4213597, 6423127757, 346212384169, 346212384169, 6423127757, 4213597, 877, 4140, 190899322

Offset: 1

R. H. Hardin, Feb 22 2012

Equivalently, the number of colorings in the rhombic hexagonal square grid graph RH_(n,k) using any number of colors up to permutation of the colors. - Andrew Howroyd, Jun 25 2017

			Table starts
...1.........1.............2................5................15
...1.........2............15..............203..............4140
...2........15...........716............83440..........18171918
...5.......203.........83440........112073062......346212384169
..15......4140......18171918.....346212384169.18633407199331522
..52....115975....6423127757.2043836452962923
.203...4213597.3376465219485
.877.190899322
...
Some solutions for n=4 k=3
..0..1..0....0..1..0....0..1..0....0..1..0....0..1..2....0..1..0....0..1..0
..2..3..1....2..3..4....2..3..2....2..3..1....2..3..0....2..3..1....2..3..2
..4..2..4....0..5..0....0..4..0....0..4..5....4..5..3....4..5..3....0..1..4
..0..5..0....1..2..1....1..2..1....5..3..4....0..1..0....0..6..4....2..0..1

Andrew Howroyd, Table of n, a(n) for n = 1..231 (terms 1..49 from R. H. Hardin)

Columns 1-5 are A000110(n-1), A020557(n-1), A208051, A208052, A208053.

Cf. A207868, A212162, A212163, A208050, A068271.

A068271 1/4 the number of colorings of an n X n rhombic hexagonal array with 4 colors.

Original entry on oeis.org

1, 12, 264, 11424, 1008576, 184910592, 71033971200, 57469424744448, 98237339264864256, 355574469749489123328, 2729407814499050197254144, 44482040254775494064841818112, 1540473331004371306422199656382464, 113440401780206156918876627438624833536

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 1..16

Cf. A068239-A068305, A000332, A002417, A027441, A212162, A212163.

a(9) from Alois P. Heinz, May 02 2012

a(10)-a(14) from Andrew Howroyd, Jun 25 2017

A068244 1/6 the number of colorings of a 3 X 3 rhombic- or staggered- hexagonal array with n colors.

Original entry on oeis.org

1, 176, 5490, 65600, 455875, 2239776, 8647716, 27962880, 78920325, 200002000, 464447126, 1003294656, 2039332295, 3935444800, 7261533000, 12884914176, 22089914121, 36733221360, 59442494650, 93866696000, 144987663051, 219503536736, 326295822700, 476993088000

Offset: 3

R. H. Hardin, Feb 24 2002

nonn

Numbers for rhombic- and staggered- hexagonal arrays differ above 4 X 4.

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

Cf. A068239-A068305, A000332, A002417, A027441, A212162, A212163.

a:= n-> (248 +(-1012 +(1786 +(-1791 +(1120 +(-448 +(112 +(-16+n)*n) *n) *n) *n) *n) *n) *n) *n/6:
seq(a(n), n=3..40);  #  Alois P. Heinz, May 02 2012

From Alois P. Heinz, May 02 2012: (Start)
G.f.: (1089*x^6+10934*x^5+26015*x^4+18500*x^3+3775*x^2+166*x+1) / (x-1)^10*x^3.
a(n) = (n^9 -16*n^8 +112*n^7 -448*n^6 +1120*n^5 -1791*n^4 +1786*n^3 -1012*n^2 +248*n)/6.  (End)

A068245 1/6 the number of colorings of a 4 X 4 rhombic- or staggered- hexagonal array with n colors.

Original entry on oeis.org

1, 7616, 5141250, 552093440, 20631905875, 395001645696, 4771909547076, 41190314035200, 275192443300005, 1502690499112000, 6971521964029766, 28275884687022336, 102456840191225975, 337289521082456320, 1022310183284613000, 2883605488481550336, 7636012822945480521

Offset: 3

R. H. Hardin, Feb 24 2002

Numbers for rhombic- and staggered- hexagonal arrays differ above 4 X 4.

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

Cf. A068239-A068305, A000332, A002417, A027441, A212162, A212163.

[(n^11 -26*n^10 +310*n^9 -2240*n^8 +10915*n^7 -37726*n^6 +94576*n^5 -172395*n^4 +224588*n^3 -199854*n^2 +109788*n -28340)*n *(n-1)*(n-2)^3/6: n in [3..19]]; // Bruno Berselli, May 03 2012

a:= n-> (-226720+ (1445104+ (-4304712+ (7968348+ (-10265148+ (9755858+ (-7068408+ (3975561+ (-1749715+ (602408+ (-160859+ (32703+ (-4898+ (510+ (-33+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 02 2012

From Alois P. Heinz, May 02 2012: (Start)
G.f.: -(7926831*x^13 +710120929*x^12 +16477733814*x^11 +144915014346*x^10 +569769493505*x^9 +1086745824783*x^8 +1040642122932*x^7 +499586289612*x^6 +115866023553*x^5 +11940350895*x^4 +465727286*x^3 +5011914*x^2 +7599*x+1) *x^3 / (x-1)^17.
a(n) = (n^16 -33*n^15 +510*n^14 -4898*n^13 +32703*n^12 -160859*n^11 +602408*n^10 -1749715*n^9  +3975561*n^8 -7068408*n^7 +9755858*n^6 -10265148*n^5 +7968348*n^4 -4304712*n^3 +1445104*n^2 -226720*n)/6. (End)

Extended beyond a(15) by Alois P. Heinz, May 02 2012

A068246 1/6 the number of colorings of a 5 X 5 rhombic hexagonal array with n colors.

Original entry on oeis.org

1, 672384, 24673292910, 47694893373440, 16222878355401375, 1842996126472816896, 98798500424990038764, 3068393771393664491520, 62960689342002146953005, 933100311834971308336000, 10639781338324232990590266, 97779035968707368095801344, 750090455889142956720814955

Offset: 3

R. H. Hardin, Feb 24 2002

nonn

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

Cf. A068239-A068305, A000332, A002417, A027441, A212162, A212163.

a:= n-> (3008737472+ (-26856982336+ (115567646848+ (-319382723824+ (636837385892+ (-975405045160+ (1192546680096+ (-1193738274422+ (995467197535+ (-699933854941+ (418375982241+ (-213720456031+ (93568827565+ (-35133626327+ (11298632622+
(-3101089711+ (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 02 2012

G.f.: (1155805517421*x^22 +898154715023598*x^21 +153334491715682431*x^20 +9260621966248364140*x^19 +250086793798293779695*x^18 +3463005755473293705486*x^17 +26809839147864527991573*x^16 +122805799859998392511056*x^15 +345417237429621912129330*x^14 +610511151468783633149340*x^13 +686259871966584143669766*x^12 +491767778082675626596168*x^11 +223082415423639038320846*x^10 +62970879259692393145420*x^9 +10739574336476388551610*x^8 +1057138433525073018576*x^7 +56029398700931117553*x^6 +1436637989069258166*x^5 +14990828199704235*x^4 +47053606279980*x^3 +24655811251*x^2+672358*x+1)*x^3 / (x-1)^26. - Alois P. Heinz, May 02 2012

Extended beyond a(10) by Alois P. Heinz, May 02 2012

A068247 1/6 the number of colorings of a 6 X 6 rhombic hexagonal array with n colors.

Original entry on oeis.org

1, 123273728, 606966551329230, 42294602754348892160, 221621345837018832499375, 227499859288036192921814016, 76749554391225308000690033388, 11559255542512176814494743592960, 945121787128699657370828476884045, 47886238054762718784097603771840000

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 A068247

Cf. A068239-A068305, A000332, A002417, A027441, A212162, A212163.

a(n) = A212163(n,6)/6. - Alois P. Heinz, Dec 20 2012

Extended beyond a(7) by Alois P. Heinz, May 02 2012

A068272 1/5 the number of colorings of an n X n rhombic hexagonal array with 5 colors.

Original entry on oeis.org

1, 36, 6588, 6169500, 29607951492, 728359861595076, 91850117930957987604, 59375832328440271451653884, 196759041550650648492160100067468, 3342365125469959463321258123750822068332, 291049545090789051667036543236086657649935910108

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, A212162, A212163.

a(8)-a(9) from Alois P. Heinz, May 02 2012

a(10)-a(11) from Andrew Howroyd, Jun 25 2017

cp's OEIS Frontend

A212162 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the rhombic hexagonal square grid graph RH_(n,n), highest powers first.

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A212195 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the staggered hexagonal square grid graph SH_(k,k).

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A208050 T(n,k)=Number of nXk 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

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A208054 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 or antidiagonal neighbor (colorings ignoring permutations of colors).

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A068271 1/4 the number of colorings of an n X n rhombic hexagonal array with 4 colors.

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A068244 1/6 the number of colorings of a 3 X 3 rhombic- or staggered- hexagonal array with n colors.

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Maple

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A068245 1/6 the number of colorings of a 4 X 4 rhombic- or staggered- hexagonal array with n colors.

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Magma

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A068246 1/6 the number of colorings of a 5 X 5 rhombic hexagonal array with n colors.

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Maple

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A068247 1/6 the number of colorings of a 6 X 6 rhombic hexagonal array with n colors.

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