A302515 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 4 or 5 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

1, 2, 2, 3, 3, 4, 5, 3, 4, 8, 8, 5, 11, 6, 16, 13, 7, 15, 9, 9, 32, 21, 13, 21, 28, 14, 14, 64, 34, 23, 52, 36, 48, 21, 22, 128, 55, 37, 118, 80, 90, 89, 28, 35, 256, 89, 63, 220, 235, 199, 184, 163, 37, 56, 512, 144, 109, 408, 541, 689, 458, 376, 297, 51, 90, 1024, 233, 183

Offset: 1

Author

R. H. Hardin, Apr 09 2018

Keywords

• nonn
• tabl

Comments

Table starts

...1..2..3...5....8...13....21.....34......55......89......144.......233

...2..3..3...5....7...13....23.....37......63.....109......183.......309

...4..4.11..15...21...52...118....220.....408.....852.....1764......3460

...8..6..9..28...36...80...235....541....1115....2554.....6095.....13920

..16..9.14..48...90..199...689...2125....5410...13908....39850....114503

..32.14.21..89..184..458..1784...7182...22544...67096...220654....775150

..64.22.28.163..376.1088..4558..23944...95681..344525..1302832...5550086

.128.35.37.297..832.2651.12324..82857..414880.1775176..7735877..39371229

.256.56.51.544.1744.6257.32336.282857.1748514.8778929.44362463.272701915

Examples

			Some solutions for n=5 k=4
..0..0..1..0. .0..1..0..1. .0..1..1..1. .0..1..0..1. .0..1..0..1
..1..1..1..0. .0..1..0..1. .0..1..0..1. .0..1..1..1. .0..1..0..1
..1..0..1..0. .0..0..1..1. .0..1..0..1. .0..0..0..1. .0..1..0..1
..1..0..1..0. .1..1..0..0. .0..1..0..1. .0..1..0..1. .0..1..0..1
..1..0..0..0. .1..0..1..0. .0..0..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 A003227(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-4) for n>7

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

k=5: a(n) = a(n-1) +6*a(n-3) +2*a(n-5) -12*a(n-6) -4*a(n-7) +8*a(n-9) for n>11

k=6: a(n) = a(n-1) +6*a(n-3) +5*a(n-4) +3*a(n-5) -8*a(n-6) -6*a(n-7) -3*a(n-8) for n>12

k=7: [order 15] for n>21

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) +2*a(n-3) +4*a(n-4) for n>7

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

n=5: [order 13] for n>17

n=6: [order 23] for n>29

n=7: [order 50] for n>55