A302010 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2 or 3 horizontally or antidiagonally adjacent elements, with upper left element zero.

1, 2, 2, 4, 8, 4, 8, 32, 32, 8, 16, 128, 240, 128, 16, 32, 512, 1808, 1808, 512, 32, 64, 2048, 13616, 25808, 13616, 2048, 64, 128, 8192, 102544, 369040, 368144, 102544, 8192, 128, 256, 32768, 772272, 5276816, 9989376, 5251712, 772272, 32768, 256, 512, 131072

Offset

1,2

Comments

Table starts

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

...2......8.......32.........128...........512.............2048

...4.....32......240........1808.........13616...........102544

...8....128.....1808.......25808........369040..........5276816

..16....512....13616......368144.......9989376........270990144

..32...2048...102544.....5251712.....270422672......13918667808

..64...8192...772272....74917424....7320574992.....714887543376

.128..32768..5816080..1068722240..198174358400...36717919842624

.256.131072.43801648.15245681888.5364752820144.1885898831169344

Links

R. H. Hardin, Table of n, a(n) for n = 1..479

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) +4*a(n-2)

k=4: a(n) = 13*a(n-1) +18*a(n-2) +a(n-3) -4*a(n-4)

k=5: a(n) = 24*a(n-1) +82*a(n-2) +34*a(n-3) -90*a(n-4) -40*a(n-5) +37*a(n-6)

k=6: [order 10] for n>12

k=7: [order 17] for n>19

Empirical for row n:

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

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

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

n=4: a(n) = 13*a(n-1) +20*a(n-2) -16*a(n-3) -64*a(n-4) for n>6

n=5: [order 12] for n>15

n=6: [order 32] for n>36

n=7: [order 78] for n>83

Example

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

Crossrefs

Column 1 and row 1 are A000079(n-1).

Column 2 and row 2 are A004171(n-1).

Column 3 and row 3 are A301779.

Sequence in context: A300804 A303456 A301443 * A301784 A316808 A299654

Adjacent sequences: A302007 A302008 A302009 * A302011 A302012 A302013

Keyword

nonn,tabl

Author

R. H. Hardin, Mar 30 2018

Status

approved