A302081 T(n,k) = Number of n X k 0..1 arrays with every element equal to 0, 1, 3 or 4 horizontally or antidiagonally adjacent elements, with upper left element zero.

1, 2, 2, 3, 5, 4, 5, 10, 13, 8, 8, 18, 33, 34, 16, 13, 35, 70, 115, 89, 32, 21, 74, 154, 265, 386, 233, 64, 34, 154, 433, 692, 1018, 1323, 610, 128, 55, 317, 1166, 2838, 3048, 3914, 4485, 1597, 256, 89, 658, 3153, 10526, 18492, 13712, 15017, 15290, 4181, 512, 144

Offset

1,2

Comments

Table starts

...1....2.....3......5.......8.......13........21.........34...........55

...2....5....10.....18......35.......74.......154........317..........658

...4...13....33.....70.....154......433......1166.......3153.........8468

...8...34...115....265.....692.....2838.....10526......36080.......126064

..16...89...386...1018....3048....18492.....94846.....421570......1921284

..32..233..1323...3914...13712...124046....893236....5075048.....30184552

..64..610..4485..15017...61536...829709...8320747...61236964....475553912

.128.1597.15290..57850..277248..5574946..78493707..745344973...7586538720

.256.4181.51977.222146.1248512.37461858.737846612.9066111545.120905954176

Links

R. H. Hardin, Table of n, a(n) for n = 1..310

Formula

Empirical for column k:

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

k=2: a(n) = 3*a(n-1) -a(n-2).

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

k=4: a(n) = a(n-1) +14*a(n-2) -a(n-3) -42*a(n-4) -a(n-5) +14*a(n-6) +a(n-7) -a(n-8).

k=5: [order 7] for n>9.

k=6: [order 32] for n>33.

k=7: [order 52] for n>54.

Empirical for row n:

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

n=2: a(n) = 2*a(n-1) +a(n-3) -a(n-4) -2*a(n-6) +a(n-7).

n=3: [order 26] for n>27.

n=4: [order 92] for n>93.

Example

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

Crossrefs

Column 1 is A000079(n-1).

Column 2 is A001519(n+1).

Row 1 is A000045(n+1).

Row 2 is A301885.

Sequence in context: A301964 A334043 A301884 * A209147 A355059 A327267

Adjacent sequences: A302078 A302079 A302080 * A302082 A302083 A302084

Keyword

nonn,tabl

Author

R. H. Hardin, Mar 31 2018

Status

approved