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

1,2

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

Links

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

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

Example

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

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.

Sequence in context: A119446 A302680 A302515 * A349122 A076918 A162620

Adjacent sequences: A303311 A303312 A303313 * A303315 A303316 A303317

Keyword

nonn,tabl

Author

R. H. Hardin, Apr 21 2018

Status

approved