A228754 T(n,k)=Number of nXk binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

1, 1, 2, 2, 3, 4, 3, 9, 8, 8, 5, 20, 39, 21, 16, 8, 50, 126, 168, 55, 32, 13, 119, 482, 780, 723, 144, 64, 21, 289, 1712, 4599, 4808, 3111, 377, 128, 34, 696, 6277, 24246, 43862, 29608, 13386, 987, 256, 55, 1682, 22700, 134440, 342207, 418370, 182288, 57597, 2584, 512

Offset

1,3

Comments

Table starts

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

...2....3......9......20........50.........119..........289............696

...4....8.....39.....126.......482........1712.........6277..........22700

...8...21....168.....780......4599.......24246.......134440.........728537

..16...55....723....4808.....43862......342207......2876170.......23326164

..32..144...3111...29608....418370.....4823826.....61534448......746135864

..64..377..13386..182288...3990739....67970044...1316714732....23857469157

.128..987..57597.1122240..38067290...957616341..28177227352...762713002760

.256.2584.247827.6908896.363121586.13491214832.602998827928.24382157716612

Links

R. H. Hardin, Table of n, a(n) for n = 1..1347

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) = 5*a(n-1) -3*a(n-2)

k=4: a(n) = 8*a(n-1) -12*a(n-2) +4*a(n-3)

k=5: a(n) = 13*a(n-1) -36*a(n-2) +29*a(n-3) -5*a(n-4) for n>5

k=6: a(n) = 21*a(n-1) -112*a(n-2) +217*a(n-3) -157*a(n-4) +36*a(n-5) for n>6

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

Empirical for row n:

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

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

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

n=4: [order 8]

n=5: [order 13]

n=6: [order 21]

n=7: [order 34]

Example

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

Crossrefs

Column 1 is A000079(n-1)

Column 2 is A001906

Column 3 is A095939

Row 1 is A000045

Row 2 is A097075(n+1)

Sequence in context: A324480 A164975 A253889 * A171830 A071506 A372645

Adjacent sequences: A228751 A228752 A228753 * A228755 A228756 A228757

Keyword

nonn,tabl

Author

R. H. Hardin Sep 02 2013

Status

approved