A078099 -id:A078099 - cp's OEIS frontend

A207997 T(n,k) = number of n X k 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).

1, 1, 1, 2, 3, 2, 4, 9, 9, 4, 8, 27, 41, 27, 8, 16, 81, 187, 187, 81, 16, 32, 243, 853, 1302, 853, 243, 32, 64, 729, 3891, 9075, 9075, 3891, 729, 64, 128, 2187, 17749, 63267, 96831, 63267, 17749, 2187, 128, 256, 6561, 80963, 441090, 1034073, 1034073, 441090, 80963

Offset: 1

R. H. Hardin, Feb 22 2012

Number of colorings of the grid graph P_n X P_k using a maximum of 3 colors up to permutation of the colors. - Andrew Howroyd, Jun 26 2017

			Table starts
..1....1.....2.......4.........8.........16...........32............64
..1....3.....9......27........81........243..........729..........2187
..2....9....41.....187.......853.......3891........17749.........80963
..4...27...187....1302......9075......63267.......441090.......3075255
..8...81...853....9075.....96831....1034073.....11045757.....117997043
.16..243..3891...63267...1034073...16932816....277458045....4547477370
.32..729.17749..441090..11045757..277458045...6978332618..175605187731
.64.2187.80963.3075255.117997043.4547477370.175605187731.6787438272198
...
Some solutions for n=4, k=3:
..0..1..2....0..1..0....0..1..0....0..1..2....0..1..2....0..1..2....0..1..0
..2..0..1....2..0..2....1..0..2....1..2..1....2..0..1....1..2..1....1..2..1
..0..2..0....0..1..0....2..1..0....0..1..2....0..2..0....0..1..2....2..0..2
..1..0..1....1..2..1....1..0..1....1..2..0....2..0..2....2..0..1....1..2..0

R. H. Hardin, Table of n, a(n) for n = 1..544
R. J. Mathar, Counting 2-way monotonic terrace forms over rectangular landscapes, vixra:1511.0225, eq. (33)-(35).
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Vertex Coloring
Wikipedia, Graph Coloring

Cf. A020698 (column 3), A078100 (column 4), A207994 (column 5), A207995 (column 6), A207996 (column 7).
Main diagonal is A207993.
Cf. A198715 (4 colorings), A198906 (5 colorings), A198982 (6 colorings), A198723 (7 colorings), A198914 (8 colorings), A207868 (unlimited).

2*T(n,m) = A078099(n,m) for m>1. - R. J. Mathar, Nov 23 2015

A222144 T(n,k) = number of n X k 0..4 arrays with no entry increasing mod 5 by 4 rightwards or downwards, starting with upper left zero.

Original entry on oeis.org

1, 4, 4, 16, 52, 16, 64, 676, 676, 64, 256, 8788, 28564, 8788, 256, 1024, 114244, 1206964, 1206964, 114244, 1024, 4096, 1485172, 50999956, 165770032, 50999956, 1485172, 4096, 16384, 19307236, 2154990196, 22767656980, 22767656980

Offset: 1

R. H. Hardin, Feb 09 2013

1/5 the number of 5-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017

			Table starts
.......1.............4...................16.........................64
.......4............52..................676.......................8788
......16...........676................28564....................1206964
......64..........8788..............1206964..................165770032
.....256........114244.............50999956................22767656980
....1024.......1485172...........2154990196..............3127020364012
....4096......19307236..........91058563924............429480137694664
...16384.....250994068........3847656513844..........58986884432558548
...65536....3262922884......162581749707796........8101544704688334244
..262144...42417997492.....6869850581244916.....1112705429924911477552
.1048576..551433967396...290283793189916884...152824358676750267429220
.4194304.7168641576148.12265868026121849524.20989638386627725143014812
...
Some solutions for n=3, k=4:
..0..0..1..1....0..0..0..0....0..0..0..0....0..0..0..0....0..0..1..1
..1..1..2..2....1..1..1..2....0..1..3..3....0..2..2..0....0..1..2..3
..3..4..0..0....1..3..1..3....2..2..0..1....0..2..2..2....1..4..2..3

Andrew Howroyd, Table of n, a(n) for n = 1..378 (terms 1..127 from R. H. Hardin)

Columns 1-7 are A000302(n-1), A222138, A222139, A222140, A222141, A222142, A222143.
Main diagonal is A068255.
Cf. A078099 (3 colorings), A222444 (4 colorings), A198906 (unlabeled 5 colorings), A222281 (6 colorings), A222340 (7 colorings), A222462 (8 colorings).

T(n,k) = 4 * (6*A198906(n,k) - 3*A207997(n,k) - 2) for n*k > 1. - Andrew Howroyd, Jun 27 2017

A222444 T(n,k) = number of n X k 0..3 arrays with entries increasing mod 4 by 0, 1 or 2 rightwards and downwards, starting with upper left zero.

Original entry on oeis.org

1, 3, 3, 9, 21, 9, 27, 147, 147, 27, 81, 1029, 2403, 1029, 81, 243, 7203, 39285, 39285, 7203, 243, 729, 50421, 642249, 1500183, 642249, 50421, 729, 2187, 352947, 10499787, 57289767, 57289767, 10499787, 352947, 2187, 6561, 2470629, 171655443

Offset: 1

R. H. Hardin, Feb 20 2013

1/4 the number of 4-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017

			Table starts
......1..........3...............9..................27.......................81
......3.........21.............147................1029.....................7203
......9........147............2403...............39285...................642249
.....27.......1029...........39285.............1500183.................57289767
.....81.......7203..........642249............57289767...............5110723191
....243......50421........10499787..........2187822609.............455924913093
....729.....352947.......171655443.........83550197745...........40672916404629
...2187....2470629......2806303725.......3190677470643.........3628419487925547
...6561...17294403.....45878770089.....121847980727187.......323690312271131451
..19683..121060821....750047661027....4653221950068669.....28876324830999722133
..59049..847425747..12262131106083..177700725073710285...2576049100980154511889
.177147.5931980229.200467073061765.6786168386579878383.229808641254065144560647
...
Some solutions for n=3, k=4:
..0..0..0..2....0..0..2..0....0..2..0..0....0..2..0..2....0..0..2..3
..1..2..2..3....0..2..3..1....2..2..2..0....0..0..0..2....0..2..3..1
..2..2..3..1....2..0..1..3....2..2..0..0....2..0..1..3....1..2..0..1

Andrew Howroyd, Table of n, a(n) for n = 1..496 (terms 1..180 from R. H. Hardin)
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Vertex Coloring
Wikipedia, Graph Coloring

Columns 1-7 are A000244(n-1), A169634(n-1), A222439, A222440, A222441, A222442, A222443.
Main diagonal is A068254.
Cf. A078099 (3 colorings), A198715 (unlabeled 4 colorings), A222144 (5 colorings), A222281 (6 colorings), A222340 (7 colorings), A222462 (8 colorings).

T(n,k) = 6*A198715(n,k) - 3 for n*k>1. - Andrew Howroyd, Jun 27 2017
Empirical for column k:
k=1: a(n) = 3*a(n-1).
k=2: a(n) = 7*a(n-1).
k=3: a(n) = 18*a(n-1) - 27*a(n-2).
k=4: a(n) = 45*a(n-1) - 267*a(n-2) + 263*a(n-3).
k=5: a(n) = 118*a(n-1) - 2811*a(n-2) + 22255*a(n-3) - 53860*a(n-4) - 54747*a(n-5) + 269406*a(n-6) - 175392*a(n-7).
k=6: [order 13]
k=7: [order 32]

A222281 T(n,k) = number of n X k 0..5 arrays with no entry increasing mod 6 by 5 rightwards or downwards, starting with upper left zero.

Original entry on oeis.org

1, 5, 5, 25, 105, 25, 125, 2205, 2205, 125, 625, 46305, 194485, 46305, 625, 3125, 972405, 17153945, 17153945, 972405, 3125, 15625, 20420505, 1513010465, 6354787485, 1513010465, 20420505, 15625, 78125, 428830605, 133450391205

Offset: 1

R. H. Hardin, Feb 14 2013

1/6 the number of 6-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017

			Table starts
........1................5......................25..........................125
........5..............105....................2205........................46305
.......25.............2205..................194485.....................17153945
......125............46305................17153945...................6354787485
......625...........972405..............1513010465................2354171487645
.....3125.........20420505............133450391205..............872117822449905
....15625........428830605..........11770577485085...........323081602357856985
....78125.......9005442705........1038187247574145........119687637492011211885
...390625.....189114296805.......91570083319317865......44339047670574481807485
..1953125....3971400232905.....8076654937439905005...16425682631297501047982145
..9765625...83399404891005...712376276332499775685.6084998755694142903356375385
.48828125.1751387502711105.62832938018547611186345
...
Some solutions for n=3, k=4:
..0..0..0..0....0..0..0..0....0..0..0..0....0..3..0..0....0..0..0..0
..4..2..0..1....1..2..0..4....0..0..0..1....0..0..3..1....0..2..3..0
..0..4..1..4....1..4..1..2....3..4..4..1....3..0..4..4....4..5..1..3

Andrew Howroyd, Table of n, a(n) for n = 1..325 (terms 1..111 from R. H. Hardin)

Columns 1-7 are A000351(n-1), 5*A009965(n-1), A222276, A222277, A222278, A222279, A222280.
Main diagonal is A068256.
Cf. A078099 (3 colorings), A222444 (4 colorings), A222144 (5 colorings), A198982 (unlabeled 6 colorings), A222340 (7 colorings), A222462 (8 colorings).

T(n, k) = 5 * (24*A198982(n,k) - 12*A198715(n,k) - 8*A207997(n,k) - 3) for n*k > 1. - Andrew Howroyd, Jun 27 2017

A222340 T(n,k) = number of n X k 0..6 arrays with no entry increasing mod 7 by 6 rightwards or downwards, starting with upper left zero.

Original entry on oeis.org

1, 6, 6, 36, 186, 36, 216, 5766, 5766, 216, 1296, 178746, 923526, 178746, 1296, 7776, 5541126, 147918906, 147918906, 5541126, 7776, 46656, 171774906, 23691810366, 122408393436, 23691810366, 171774906, 46656, 279936, 5325022086

Offset: 1

R. H. Hardin, Feb 15 2013

1/7 the number of 7-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017

			Table starts
.......1.............6...................36........................216
.......6...........186.................5766.....................178746
......36..........5766...............923526..................147918906
.....216........178746............147918906...............122408393436
....1296.......5541126..........23691810366............101297497221786
....7776.....171774906........3794659477146..........83827445649884946
...46656....5325022086......607781352505806.......69370328359709445996
..279936..165075684666....97346856728146986....57406526220963704077986
.1679616.5117346224646.15591808593304758846.47506035082750189614687546
...
Some solutions for n=3, k=4:
..0..0..2..0....0..2..2..0....0..0..0..0....0..2..0..0....0..2..0..0
..0..5..3..0....0..2..5..0....0..1..5..0....0..5..0..0....0..0..5..0
..3..1..4..5....4..4..2..3....3..6..1..4....2..2..2..2....4..1..3..4

Andrew Howroyd, Table of n, a(n) for n = 1..325 (terms 1..84 from R. H. Hardin)

Columns 1-6 are A000400(n-1), A222335, A222336, A222337, A222338, A222339.
Main diagonal is A068257.
Cf. A078099 (3 colorings), A222444 (4 colorings), A222144 (5 colorings), A222281 (6 colorings), A198723 (unlabeled 7 colorings), A222462 (8 colorings).

T(n, k) = 6 * (120*A198723(n,k) - 60*A198906(n,k) - 40*A198715(n,k) - 15*A207997(n,k) - 4) for n*k > 1. - Andrew Howroyd, Jun 27 2017

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.

Original entry on oeis.org

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).

A020698 a(n) = 5a(n-1) - 2a(n-2), with a(0)=2, a(1)=9.

Original entry on oeis.org

2, 9, 41, 187, 853, 3891, 17749, 80963, 369317, 1684659, 7684661, 35053987, 159900613, 729395091, 3327174229, 15177080963, 69231056357, 315801119859, 1440543486581, 6571115193187, 29974488992773, 136730214577491, 623702094901909, 2845050045354563

Offset: 0

David W. Wilson

Coincides with Pisot sequence L(2,9) (at least for first 1000 terms).
Coincides with Pisot sequence E(2,9) (at least for first 1000 terms).
Theorem: E(2,9) satisfies a(n) = 5 a(n - 1) 2 2 a(n - 2) for n>=2. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the conjecture. - N. J. A. Sloane, Sep 09 2016
Number of ways to 3-color a 3 X (n+1) rectangular grid ignoring permutations of the colors. - Andrew Woods, Sep 07 2011

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. B. Ekhad, N. J. A. Sloane, D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT] (2016)
Index entries for linear recurrences with constant coefficients, signature (5,-2).

See A008776 for definitions of Pisot sequences.

Cf. A078099.

m:=24; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((2-x)/(1-5*x+2*x^2))); // Bruno Berselli, Sep 06 2011

I:=[2, 9]; [n le 2 select I[n] else 5*Self(n-1)-2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 19 2013

LinearRecurrence[{5,-2},{2,9},30] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
CoefficientList[Series[(2 - x)/(1 - 5 x + 2 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 19 2013 *)

PARI
```
a(n)=([2,1,2;1,1,1;2,1,2]^(n+1))[1,3]
```

a(k-1) = [M^k]1,3, where M is the 3 X 3 matrix [2,1,2; 1,1,1; 2,1,2]. - _Simone Severini, Jun 12 2006
If p[i]=Fibonacci(2i+1) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010
From Bruno Berselli, Sep 06 2011: (Start)
G.f.: (2-x)/(1-5*x+2*x^2).
a(n) = ((17+4*sqrt(17))*(5+sqrt(17))^n+(17-4*sqrt(17))*(5-sqrt(17))^n)/(17*2^n).
a(-n)*2^n = A052984(n-2). (End)
E.g.f.: 2*exp(5*x/2)*(17*cosh(sqrt(17)*x/2) + 4*sqrt(17)*sinh(sqrt(17)*x/2))/17. - Stefano Spezia, Jun 17 2025

A068253 1/3 of the number of colorings of an n X n square array with 3 colors.

Original entry on oeis.org

1, 6, 82, 2604, 193662, 33865632, 13956665236, 13574876544396, 31191658416342674, 169426507164530254380, 2176592549084872196370724, 66158464020552857153017287240, 4759146677426447759184119036493676, 810410082813497381147177065840601910384

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..24

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.
See A047938 for number of improper colorings.
Main diagonal of A078099.
Twice A207993 for n>1.

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, n]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Nov 01 2017, after code from A078099 *)

For formula see A078099.

More terms from Vladeta Jovovic, Jul 22 2004
a(11)-a(12) from Alois P. Heinz, Mar 25 2009
a(13)-a(14) from Andrew Howroyd, Jun 26 2017

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

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

cp's OEIS Frontend

A207997 T(n,k) = number of n X k 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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A222144 T(n,k) = number of n X k 0..4 arrays with no entry increasing mod 5 by 4 rightwards or downwards, starting with upper left zero.

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A222444 T(n,k) = number of n X k 0..3 arrays with entries increasing mod 4 by 0, 1 or 2 rightwards and downwards, starting with upper left zero.

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A222281 T(n,k) = number of n X k 0..5 arrays with no entry increasing mod 6 by 5 rightwards or downwards, starting with upper left zero.

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A222340 T(n,k) = number of n X k 0..6 arrays with no entry increasing mod 7 by 6 rightwards or downwards, starting with upper left zero.

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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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A020698 a(n) = 5*a(n-1) - 2*a(n-2), with a(0)=2, a(1)=9.

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Magma

Magma

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A068253 1/3 of the number of colorings of an n X n square array with 3 colors.

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

A020698 a(n) = 5a(n-1) - 2a(n-2), with a(0)=2, a(1)=9.