A068255 -id:A068255 - cp's OEIS frontend

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).

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

Offset: 1

Alois P. Heinz, Apr 27 2012

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.

			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, ...

Eric Weisstein's World of Mathematics, Grid Graph
Wikipedia, Chromatic Polynomial

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.

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

cp's OEIS Frontend

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).

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