A303421 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 5 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

1, 2, 2, 4, 8, 4, 8, 32, 32, 8, 16, 128, 252, 128, 16, 32, 512, 1985, 1988, 512, 32, 64, 2048, 15647, 30897, 15684, 2048, 64, 128, 8192, 123337, 480953, 480960, 123732, 8192, 128, 256, 32768, 972168, 7486281, 14783632, 7486369, 976132, 32768, 256, 512, 131072

Offset

1,2

Comments

Table starts

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

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

...4.....32......252........1985..........15647...........123337

...8....128.....1988.......30897.........480953..........7486281

..16....512....15684......480960.......14783632........454377792

..32...2048...123732.....7486369......454381369......27575294129

..64...8192...976132...116529645....13965759339....1673515027797

.128..32768..7700788..1813851698...429248347970..101563813268522

.256.131072.60752164.28233652317.13193275586412.6163796529251277

Links

R. H. Hardin, Table of n, a(n) for n = 1..311

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) +6*a(n-2) +8*a(n-3)

k=4: a(n) = 14*a(n-1) +21*a(n-2) +54*a(n-3) -23*a(n-4) -18*a(n-5) -20*a(n-6)

k=5: [order 16]

k=6: [order 31]

k=7: [order 58]

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) +21*a(n-3) +18*a(n-4)

n=4: [order 9]

n=5: [order 19]

n=6: [order 44]

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..1
..0..1..0..0. .1..1..1..0. .0..1..0..1. .1..0..1..1. .0..0..0..1
..1..1..1..0. .0..0..1..0. .0..1..0..1. .0..0..0..0. .1..0..0..0
..0..0..1..0. .0..1..0..1. .0..0..0..1. .1..1..1..0. .0..1..0..1
..1..0..0..0. .1..1..0..0. .0..1..1..0. .0..0..0..0. .0..0..1..1

Crossrefs

Column 1 is A000079(n-1).

Column 2 is A004171(n-1).

Row 1 is A000079(n-1).

Row 2 is A004171(n-1).

Sequence in context: A299654 A317525 A300208 * A301407 A213418 A317517

Adjacent sequences: A303418 A303419 A303420 * A303422 A303423 A303424

Keyword

nonn,tabl

Author

R. H. Hardin, Apr 23 2018

Status

approved