A297299 T(n,k)=Number of nXk 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 5, 5, 1, 1, 1, 6, 9, 10, 8, 1, 1, 1, 9, 17, 21, 19, 13, 1, 1, 1, 13, 32, 50, 49, 36, 21, 1, 1, 1, 19, 60, 130, 157, 114, 69, 34, 1, 1, 1, 28, 113, 332, 600, 495, 266, 131, 55, 1, 1, 1, 41, 213, 840, 2161, 2816, 1574, 620, 250, 89, 1, 1, 1

Offset: 1

Author

R. H. Hardin, Dec 27 2017

Keywords

• nonn
• tabl

Comments

Table starts

.1.1..1...1....1.....1......1.......1........1.........1..........1...........1

.1.1..2...3....4.....6......9......13.......19........28.........41..........60

.1.1..3...5....9....17.....32......60......113.......213........401.........755

.1.1..5..10...21....50....130.....332......840......2128.......5408.......13772

.1.1..8..19...49...157....600....2161.....7479.....25639......88307......306281

.1.1.13..36..114...495...2816...14138....64406....288079....1310914.....6040052

.1.1.21..69..266..1574..13504...93906...551477...3159693...18792129...113862870

.1.1.34.131..620..5031..65521..629563..4738693..34652662..269300602..2142898994

.1.1.55.250.1446.16123.319835.4235806.40753529.379882818.3857849537.40317718384

Examples

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

Links

• R. H. Hardin, Table of n, a(n) for n = 1..680

Crossrefs

Column 3 is A000045(n+1).

Column 4 is A158943(n+1).

Row 2 is A000930.

Formula

Empirical for column k:

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

k=2: a(n) = a(n-1)

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

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

k=5: a(n) = 2*a(n-1) +2*a(n-2) -2*a(n-3) -2*a(n-4) for n>5

k=6: a(n) = 4*a(n-1) -8*a(n-3) -2*a(n-4) +4*a(n-5) +4*a(n-6) +a(n-7) -a(n-8)

k=7: [order 12] for n>14

Empirical for row n:

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

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

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

n=4: [order 13] for n>16

n=5: [order 32] for n>37

n=6: [order 68] for n>76