A297395 T(n,k)=Number of nXk 0..1 arrays with every 1 horizontally, diagonally or antidiagonally adjacent to 1 neighboring 1.

1, 2, 1, 3, 5, 1, 4, 9, 9, 1, 6, 13, 19, 20, 1, 9, 33, 37, 57, 41, 1, 13, 69, 127, 126, 139, 85, 1, 19, 121, 323, 700, 385, 369, 178, 1, 28, 253, 763, 2569, 3175, 1243, 963, 369, 1, 41, 529, 2121, 7779, 14940, 15541, 3924, 2489, 769, 1, 60, 1013, 5557, 31081, 58901, 99682

Offset: 1

Author

R. H. Hardin, Dec 29 2017

Keywords

• nonn
• tabl

Comments

Table starts

.1...2....3.....4.......6........9........13.........19...........28

.1...5....9....13......33.......69.......121........253..........529

.1...9...19....37.....127......323.......763.......2121.........5557

.1..20...57...126.....700.....2569......7779......31081.......117084

.1..41..139...385....3175....14940.....58901.....325922......1616869

.1..85..369..1243...15541....99682....514945....3977868.....27131403

.1.178..963..3924...74736...640562...4279111...46261441....428200086

.1.369.2489.12477..358341..4101278..35870939..540319235...6780786267

.1.769.6523.39625.1729617.26607999.302197213.6362528482.108762242579

Examples

			Some solutions for n=5 k=4
..0..0..1..1. .0..1..1..0. .0..1..0..0. .0..0..1..1. .0..0..0..0
..0..0..0..0. .0..0..0..0. .1..0..0..0. .0..0..0..0. .0..0..1..0
..0..1..1..0. .0..0..0..0. .0..0..0..0. .0..0..1..0. .0..0..0..1
..0..0..0..0. .0..0..0..1. .0..0..0..0. .0..0..0..1. .0..1..0..0
..0..0..1..1. .0..0..1..0. .1..1..0..0. .0..0..0..0. .1..0..0..0
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..420

Crossrefs

Column 2 is A105309(n+1).

Row 1 is A000930(n+1).

Row 2 is A089977(n+1).

Formula

Empirical for column k:

k=1: a(n) = a(n-1)

k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3) -a(n-4)

k=3: a(n) = a(n-1) +4*a(n-2) +2*a(n-3) -4*a(n-4)

k=4: a(n) = a(n-1) +6*a(n-2) +4*a(n-3) -3*a(n-4) -a(n-5) -2*a(n-6) -a(n-7)

k=5: [order 20]

k=6: [order 25]

k=7: [order 55]

Empirical for row n:

n=1: a(n) = a(n-1) +a(n-3)

n=2: a(n) = a(n-1) +4*a(n-3)

n=3: a(n) = a(n-1) +a(n-2) +8*a(n-3) +a(n-4) -2*a(n-6)

n=4: [order 8]

n=5: [order 21]

n=6: [order 31]

n=7: [order 69]