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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. 14
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
1/4 the number of 4-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017
LINKS
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
FORMULA
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]
EXAMPLE
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
CROSSREFS
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).
Sequence in context: A099465 A099094 A222169 * A206492 A007683 A316220
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Feb 20 2013
STATUS
approved

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)