A068255 -id:A068255

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A182406

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

+10
28

1, 0, 2, 0, 2, 3, 0, 2, 18, 4, 0, 2, 246, 84, 5, 0, 2, 7812, 9612, 260, 6, 0, 2, 580986, 6000732, 142820, 630, 7, 0, 2, 101596896, 20442892764, 828850160, 1166910, 1302, 8, 0, 2, 41869995708, 380053267505964, 50820390410180, 38128724910, 6464682, 2408, 9

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

The square grid graph G_(n,n) has n^2 = A000290(n) vertices and 2*n*(n-1) = A046092(n-1) edges. The chromatic polynomial of G_(n,n) has n^2+1 = A002522(n) coefficients.

LINKS

Table of n, a(n) for n=1..45.

Eric Weisstein's World of Mathematics, Grid Graph

Wikipedia, Chromatic Polynomial

EXAMPLE

Square array A(n,k) begins:

1, 0, 0, 0, 0, ...

2, 2, 2, 2, 2, ...

3, 18, 246, 7812, 580986, ...

4, 84, 9612, 6000732, 20442892764, ...

5, 260, 142820, 828850160, 50820390410180, ...

6, 630, 1166910, 38128724910, 21977869327169310, ...

CROSSREFS

Columns k=1-7 give: A000027, A091940, A068239*2, A068240*2, A068241*2, A068242*2, A068243*2.

Rows n=1-20 give: A000007, A007395, A068253*3, A068254*4, A068255*5, A068256*6, A068257*7, A068258*8, A068259*9, A068260*10, A068261*11, A068262*12, A068263*13, A068264*14, A068265*15, A068266*16, A068267*17, A068268*18, A068269*19, A068270*20.

Cf. A182368.

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Apr 27 2012

STATUS

approved

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.

+10
14

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

1/5 the number of 5-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..378 (terms 1..127 from R. H. Hardin)

FORMULA

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

EXAMPLE

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

CROSSREFS

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).