A303314 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 4, 5 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

1, 2, 2, 3, 3, 4, 5, 3, 4, 8, 8, 5, 12, 6, 16, 13, 7, 17, 11, 9, 32, 21, 13, 24, 36, 19, 14, 64, 34, 23, 67, 50, 74, 34, 22, 128, 55, 37, 158, 128, 139, 165, 53, 35, 256, 89, 63, 298, 439, 410, 349, 361, 83, 56, 512, 144, 109, 595, 1085, 1799, 1221, 853, 783, 136, 90, 1024, 233

Offset: 1

Author

R. H. Hardin, Apr 21 2018

Keywords

• nonn
• tabl

Comments

Table starts

...1..2...3....5....8....13.....21......34.......55........89........144

...2..3...3....5....7....13.....23......37.......63.......109........183

...4..4..12...17...24....67....158.....298......595......1337.......2863

...8..6..11...36...50...128....439....1085.....2431......6452......17455

..16..9..19...74..139...410...1799....5907....16494.....53290.....184915

..32.14..34..165..349..1221...7096...30280...102683....403872....1783894

..64.22..53..361..853..3453..26184..148313...618149...2955145...16591424

.128.35..83..783.2180.10223.100128..746323..3851318..22515378..159560449

.256.56.136.1710.5525.30247.387892.3784002.23967605.171306353.1539204838

Examples

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

Links

• R. H. Hardin, Table of n, a(n) for n = 1..511

Crossrefs

Column 1 is A000079(n-1).

Column 2 is A001611(n+1).

Row 1 is A000045(n+1).

Row 2 is A003229(n-1) for n>2.

Formula

Empirical for column k:

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

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

k=3: a(n) = a(n-1) +a(n-3) +a(n-4) for n>7

k=4: a(n) = a(n-1) +a(n-2) +3*a(n-3) +2*a(n-4) -a(n-5) -2*a(n-6) -a(n-7) for n>10

k=5: a(n) = a(n-1) +9*a(n-3) +2*a(n-4) +4*a(n-5) -10*a(n-6) -6*a(n-7) +4*a(n-9) for n>12

k=6: [order 8] for n>11

k=7: [order 20] for n>23

Empirical for row n:

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

n=2: a(n) = a(n-1) +2*a(n-3) for n>5

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

n=4: a(n) = a(n-1) +a(n-2) +5*a(n-3) +9*a(n-4) -3*a(n-5) -7*a(n-6) -2*a(n-7) for n>11

n=5: [order 12] for n>16

n=6: [order 23] for n>28

n=7: [order 46] for n>51