A301841 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1 or 2 horizontally or antidiagonally adjacent elements, with upper left element zero.

1, 2, 2, 4, 8, 4, 8, 25, 32, 8, 16, 81, 139, 128, 16, 32, 263, 678, 773, 512, 32, 64, 855, 3182, 5748, 4299, 2048, 64, 128, 2778, 15199, 39703, 48802, 23909, 8192, 128, 256, 9027, 72514, 281758, 496085, 414385, 132971, 32768, 256, 512, 29333, 346244

Offset: 1

Author

R. H. Hardin, Mar 27 2018

Keywords

• nonn
• tabl

Comments

Table starts

...1......2.......4.........8..........16...........32.............64

...2......8......25........81.........263..........855...........2778

...4.....32.....139.......678........3182........15199..........72514

...8....128.....773......5748.......39703.......281758........1986213

..16....512....4299.....48802......496085......5240684.......54948498

..32...2048...23909....414385.....6196305.....97439921.....1518341751

..64...8192..132971...3518619....77396422...1812097252....41966406867

.128..32768..739525..29877293...966770632..33701001773..1159968653556

.256.131072.4112907.253694309.12076215811.626769301255.32062561937804

Examples

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

Links

• R. H. Hardin, Table of n, a(n) for n = 1..417

Crossrefs

Column 1 is A000079(n-1).

Column 2 is A004171(n-1).

Row 1 is A000079(n-1).

Formula

Empirical for column k:

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

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

k=3: a(n) = 7*a(n-1) -8*a(n-2) for n>3

k=4: a(n) = 13*a(n-1) -46*a(n-2) +72*a(n-3) -57*a(n-4) +16*a(n-5) for n>6

k=5: [order 11] for n>13

k=6: [order 25] for n>27

k=7: [order 53] for n>56

Empirical for row n:

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

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

n=3: [order 12] for n>14

n=4: [order 35] for n>38

n=5: [order 99] for n>104