A207997 -id:A207997 - cp's OEIS frontend

A222462 T(n,k) = number of n X k 0..7 arrays with no entry increasing mod 8 by 7 rightwards or downwards, starting with upper left zero.

1, 7, 7, 49, 301, 49, 343, 12943, 12943, 343, 2401, 556549, 3418807, 556549, 2401, 16807, 23931607, 903055069, 903055069, 23931607, 16807, 117649, 1029059101, 238535974201, 1465295106499, 238535974201, 1029059101, 117649, 823543

Offset: 1

R. H. Hardin, Feb 21 2013

1/8 the number of 8-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017

			Table starts
......1.............7..................49........................343
......7...........301...............12943.....................556549
.....49.........12943.............3418807..................903055069
....343........556549...........903055069..............1465295106499
...2401......23931607........238535974201...........2377584520856755
..16807....1029059101......63007686842527........3857863258420747009
.117649...44249541343...16643060295393343.....6259760185235726701945
.823543.1902730277749.4396153388210813341.10157072698503130798653535
...
Some solutions for n=3, k=4:
..0..4..2..3....0..0..0..4....0..4..6..1....0..4..0..4....0..2..6..2
..0..0..5..6....0..0..4..6....0..0..1..5....0..0..6..0....0..0..2..3
..0..0..0..1....0..0..5..1....0..0..3..5....0..0..0..1....0..0..3..5

Andrew Howroyd, Table of n, a(n) for n = 1..276 (terms 1..83 from R. H. Hardin)

Columns 1-5 are A000420(n-1), 7*43^(n-1), A222459, A222460, A222461.
Main diagonal is A068258.
Cf. A078099 (3 colorings), A222444 (4 colorings), A222144 (5 colorings), A222281 (6 colorings), A222340 (7 colorings), A198914 (unlabeled 8 colorings).

T(n, k) = 7 * (720*A198914(n,k) - 360*A198982(n,k) - 240*A198906(n,k) - 90*A198715(n,k) - 24*A207997(n,k) - 5) for n*k > 1. - Andrew Howroyd, Jun 27 2017
Empirical for column k:
k=1: a(n) = 7*a(n-1).
k=2: a(n) = 43*a(n-1).
k=3: a(n) = 270*a(n-1) - 1547*a(n-2).
k=4: a(n) = 1689*a(n-1) - 108775*a(n-2) + 1672631*a(n-3).
k=5: a(n) = 10754*a(n-1) - 8060499*a(n-2) + 2219242223*a(n-3) - 245682627864*a(n-4) + 5798947687589*a(n-5) + 448113231493438*a(n-6) - 2763020698450992*a(n-7).

A078100 1/6 of the number of ways of 3-coloring a 4 X n grid.

Original entry on oeis.org

4, 27, 187, 1302, 9075, 63267, 441090, 3075255, 21440547, 149482638, 1042187067, 7266087315, 50658875658, 353191693599, 2462438631411, 17168025532662, 119694800484387, 834507453158019, 5818153224352338, 40563936024707079, 282810170576026755

Offset: 1

N. J. A. Sloane, Dec 05 2002

Also the number of 3-colorings of the P_4 X P_n grid graph up to permutation of the colors. - Andrew Howroyd, Jun 26 2017

Michael S. Paterson (Warwick), personal communication.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-15,6).

Row 4 of (1/2)*A078099.

Row 4 of A207997.

I:=[4,27,187]; [n le 3 select I[n] else 9*Self(n-1)-15*Self(n-2)+6*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Feb 13 2016

a:= n-> (Matrix([[27, 4, 2/3]]). Matrix([[9, 1, 0], [ -15, 0, 1], [6, 0, 0]])^n)[1, 3]: seq(a(n), n=1..30); # Alois P. Heinz, Mar 23 2009

LinearRecurrence[{9, -15, 6}, {4, 27, 187}, 21] (* Jean-François Alcover, Feb 13 2016 *)

See A078099 for formula.

G.f.: x*(9*x-4-4*x^2) / (6*x^3-15*x^2+9*x-1). - Alois P. Heinz, Mar 23 2009

More terms from Alois P. Heinz, Mar 23 2009

Name clarified by Andrew Howroyd, Jun 26 2017

A207993 Number of n X n 0..2 arrays with new values 0..2 introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 3, 41, 1302, 96831, 16932816, 6978332618, 6787438272198, 15595829208171337, 84713253582265127190, 1088296274542436098185362, 33079232010276428576508643620, 2379573338713223879592059518246838

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

Diagonal of A207997.

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

R. H. Hardin, Table of n, a(n) for n = 1..16

A207997 = Import["https://oeis.org/A207997/b207997.txt", "Table"][[All, 2]];
a[n_] := A207997[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 16}] (* Jean-François Alcover, Sep 23 2019 *)

A207994 Number of nX5 0..2 arrays with new values 0..2 introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

8, 81, 853, 9075, 96831, 1034073, 11045757, 117997043, 1260537911, 13466147569, 143857201093, 1536809621307, 16417559602831, 175386899980873, 1873638094198285, 20015860487662275, 213827138093982759

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

Column 5 of A207997.

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

R. H. Hardin, Table of n, a(n) for n = 1..210
Index entries for linear recurrences with constant coefficients, signature (16,-65,92,-48,8).

a(n) = 16*a(n-1) -65*a(n-2) +92*a(n-3) -48*a(n-4) +8*a(n-5).

G.f.: -x*(8-47*x+77*x^2-44*x^3+8*x^4) / ( -1+16*x-65*x^2+92*x^3-48*x^4+8*x^5 ). - R. J. Mathar, Nov 23 2015

A207995 Number of nX6 0..2 arrays with new values 0..2 introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

16, 243, 3891, 63267, 1034073, 16932816, 277458045, 4547477370, 74538711609, 1221819475953, 20027983390866, 328298744831580, 5381481886580865, 88213445048426316, 1445998260462433698, 23702862077090281716

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

Column 6 of A207997.

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

R. H. Hardin, Table of n, a(n) for n = 1..210
Index entries for linear recurrences with constant coefficients, signature (30,-291,1278,-2901,3519,-2152,516).

a(n) = 30*a(n-1) -291*a(n-2) +1278*a(n-3) -2901*a(n-4) +3519*a(n-5) -2152*a(n-6) +516*a(n-7).

G.f.: -x*(16-237*x+1257*x^2-3198*x^3+4206*x^4-2736*x^5+688*x^6) / ( -1+30*x-291*x^2+1278*x^3-2901*x^4+3519*x^5-2152*x^6+516*x^7 ). - R. J. Mathar, Nov 23 2015

A078101 1/6 of the number of ways of 3-coloring an (n-1) X n grid.

Original entry on oeis.org

1, 9, 187, 9075, 1034073, 277458045, 175605187731, 262459366542859, 927063711694234937, 7743238400519517700687, 152996488947929392223648350, 7153582340115101979222478030231, 791692010951982239786844983500390201, 207426783553049237691620430245372971070275

Offset: 2

N. J. A. Sloane, Dec 05 2002

nonn

Also the number of 3-colorings of the P_{n-1} X P_n grid graph up to permutation of the colors. - Andrew Howroyd, Jun 26 2017

Michael S. Paterson (Warwick), personal communication.

Andrew Howroyd, Table of n, a(n) for n = 2..24

A diagonal of A078099 and A207997.

M[1] = {{1}};
M[m_] := M[m] = {{M[m - 1], Transpose[M[m - 1]]}, {Array[0 &, {2^(m - 2), 2^(m - 2)}], M[m - 1]}} // ArrayFlatten; W[m_] := M[m] + Transpose[M[m]];
T[m_, 1] := 2^(m - 1);
T[1, n_] := 2^(n - 1);
T[m_, n_] := MatrixPower[ W[m], n - 1] // Flatten // Total;
a[n_] := T[n - 1, n]/2;
Table[Print[n]; a[n], {n, 2, 15}] (* Jean-François Alcover, Sep 16 2019 *)

See A078099 for formula.

a(n) = A207997(n-1, n) = A078099(n-1, n)/2. - Andrew Howroyd, Jun 26 2017

a(7)-a(13) from Alois P. Heinz, Mar 25 2009

Name clarified and a(14)-a(15) from Andrew Howroyd, Jun 26 2017

A207996 Number of nX7 0..2 arrays with new values 0..2 introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

32, 729, 17749, 441090, 11045757, 277458045, 6978332618, 175605187731, 4419979346851, 111261280858024, 2800819361992659, 70507189155056781, 1774944424628356230, 44682493917576202087, 1124839588414204665607

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

Column 7 of A207997.

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

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

a(n) = 55*a(n-1) -1109*a(n-2) +11330*a(n-3) -67206*a(n-4) +247404*a(n-5) -582440*a(n-6) +881876*a(n-7) -846764*a(n-8) +499200*a(n-9) -172400*a(n-10) +33152*a(n-11) -3264*a(n-12) +128*a(n-13).

G.f.: -x*(32 -1031*x +13142*x^2 -89204*x^3 +360470*x^4 -909704*x^5 +1454814*x^6 -1461492*x^7 +896144*x^8 -320568*x^9 +63472*x^10 -6400*x^11 +256*x^12) / ( -1 +55*x -1109*x^2 +11330*x^3 -67206*x^4 +247404*x^5 -582440*x^6 +881876*x^7 -846764*x^8 +499200*x^9 -172400*x^10 +33152*x^11 -3264*x^12 +128*x^13 ). - R. J. Mathar, Nov 23 2015

A078102 1/6 of the number of ways of 3-coloring an (n-2) X n grid.

Original entry on oeis.org

2, 27, 853, 63267, 11045757, 4547477370, 4419979346851, 10150938472416408, 55117503183129188479, 707887801249881516079368, 21511908182992495395699279579, 1547207013442473554135873920560606, 263429541331756165013316290711160389207

Offset: 3

N. J. A. Sloane, Dec 05 2002

nonn

Also the number of 3-colorings of the P_{n-2} X P_n grid graph up to permutation of the colors. - Andrew Howroyd, Jun 26 2017

Michael S. Paterson (Warwick), personal communication.

A diagonal of A078099 and A207997.

M[1] = {{1}}; M[m_] := M[m] = {{M[m - 1], Transpose[M[m - 1]]}, {Array[0&, {2^(m - 2), 2^(m - 2)}], M[m - 1]}} // ArrayFlatten;
W[m_] := M[m] + Transpose[M[m]];
T[m_, 1] := 2^(m - 1); T[1, n_] := 2^(n - 1); T[m_, n_] := MatrixPower[ W[m], n - 1] // Flatten // Total;
a[n_] := T[n - 2, n]/2;
Table[a[n], {n, 3, 15}] (* Jean-François Alcover, Aug 27 2019 *)

See A078099 for formula.

a(n) = A207997(n-2, n) = A078099(n-2, n)/2. - Andrew Howroyd, Jun 26 2017

a(7)-a(14) from Alois P. Heinz, Mar 24 2009

Name clarified and a(15) from Andrew Howroyd, Jun 26 2017

cp's OEIS Frontend

A222462 T(n,k) = number of n X k 0..7 arrays with no entry increasing mod 8 by 7 rightwards or downwards, starting with upper left zero.

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A078100 1/6 of the number of ways of 3-coloring a 4 X n grid.

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A207993 Number of n X n 0..2 arrays with new values 0..2 introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

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Mathematica

A207994 Number of nX5 0..2 arrays with new values 0..2 introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

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A207995 Number of nX6 0..2 arrays with new values 0..2 introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

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A078101 1/6 of the number of ways of 3-coloring an (n-1) X n grid.

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Mathematica

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A207996 Number of nX7 0..2 arrays with new values 0..2 introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

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Keywords

Comments

Examples

Links

Formula

A078102 1/6 of the number of ways of 3-coloring an (n-2) X n grid.

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Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

Formula

Extensions