A316693 T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1, 2, 6, 7 or 8 king-move adjacent elements, with upper left element zero.

1, 2, 2, 4, 4, 4, 8, 3, 3, 8, 16, 5, 8, 5, 16, 32, 8, 9, 9, 8, 32, 64, 13, 21, 23, 21, 13, 64, 128, 21, 42, 44, 44, 42, 21, 128, 256, 34, 81, 125, 88, 125, 81, 34, 256, 512, 55, 165, 313, 359, 359, 313, 165, 55, 512, 1024, 89, 330, 773, 1125, 2081, 1125, 773, 330, 89, 1024, 2048

Offset

1,2

Comments

Table starts

...1..2...4....8....16.....32......64......128.......256........512........1024

...2..4...3....5.....8.....13......21.......34........55.........89.........144

...4..3...8....9....21.....42......81......165.......330........657........1317

...8..5...9...23....44....125.....313......773......1964.......4948.......12456

..16..8..21...44....88....359....1125.....3117.....10022......31545.......95424

..32.13..42..125...359...2081....9442....38020....174483.....785604.....3427907

..64.21..81..313..1125...9442...61454...342508...2212378...14109348....86477928

.128.34.165..773..3117..38020..342508..2431114..21261766..184273462..1497908640

.256.55.330.1964.10022.174483.2212378.21261766.257262996.3099570150.34685951835

Links

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

Formula

Empirical for column k:

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

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

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

k=4: a(n) = a(n-1) +2*a(n-2) +5*a(n-3) +a(n-4) -2*a(n-5) -6*a(n-6) -4*a(n-7) for n>11

k=5: [order 7] for n>11

k=6: [order 16] for n>21

k=7: [order 46] for n>52

Example

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

Crossrefs

Column 1 is A000079(n-1).

Column 2 is A000045(n+1).

Sequence in context: A304230 A305586 A305040 * A230014 A319414 A303961

Adjacent sequences: A316690 A316691 A316692 * A316694 A316695 A316696

Keyword

nonn,tabl

Author

R. H. Hardin, Jul 10 2018

Status

approved