A222139 -id:A222139 - cp's OEIS frontend

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.

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

A355883 Number of ways to 5-color a 3 X n grid ignoring the variations of two colors.

Original entry on oeis.org

4, 169, 7141, 301741, 12749989, 538747549, 22764640981, 961914128461, 40645437426949, 1717462645311229, 72570948297479221, 3066467006530462381, 129572785291363217509, 5475065165353811151709, 231347489347123368595861, 9775529461439509493215501

Offset: 1

Gerhard Kirchner, Jul 24 2022

See A355881 for a general formula.

			a(1) = 4, 5 colors 1,2,3,4,5: 121, 123, 124, 125.
The first two colors do not vary.

Paolo Xausa, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (45,-116).

Cf. A355881, A355882.

LinearRecurrence[{45, -116}, {4, 169}, 20] (* Paolo Xausa, Oct 03 2024 *)

a(n) = A222139(n)/4.
G.f.: x*(4-11*x)/(1-45*x+116*x^2).
a(n) = 45*a(n-1) - 116*a(n-2) with a(1) = 4, a(2) = 169.
a(n) = 2^(-3-n)*((45 - sqrt(1561))^n*(11*sqrt(1561) - 433) + (45 + sqrt(1561))^n*(11*sqrt(1561) + 433))/(29*sqrt(1561)). - Stefano Spezia, Jul 24 2022

cp's OEIS Frontend

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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