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T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 6 horizontally, vertically or antidiagonally adjacent elements, with upper left element zero.
7

%I #4 Mar 22 2018 11:58:20

%S 1,2,2,3,3,3,5,2,2,5,8,3,5,3,8,13,6,15,15,6,13,21,6,45,83,45,6,21,34,

%T 11,156,329,329,156,11,34,55,26,479,1576,2803,1576,479,26,55,89,43,

%U 1455,7454,21787,21787,7454,1455,43,89,144,82,4493,35118,164346,268802,164346

%N T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 6 horizontally, vertically or antidiagonally adjacent elements, with upper left element zero.

%C Table starts

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

%C ..2..3....2......3.......6.........6..........11............26.............43

%C ..3..2....5.....15......45.......156.........479..........1455...........4493

%C ..5..3...15.....83.....329......1576........7454.........35118.........164664

%C ..8..6...45....329....2803.....21787......164346.......1261122........9663349

%C .13..6..156...1576...21787....268802.....3243936......39741611......487928765

%C .21.11..479...7454..164346...3243936....62812639....1238986634....24457707005

%C .34.26.1455..35118.1261122..39741611..1238986634...39447933621..1251613605582

%C .55.43.4493.164664.9663349.487928765.24457707005.1251613605582.63780972381537

%H R. H. Hardin, <a href="/A301498/b301498.txt">Table of n, a(n) for n = 1..219</a>

%F Empirical for column k:

%F k=1: a(n) = a(n-1) +a(n-2)

%F k=2: a(n) = 2*a(n-1) -a(n-2) +4*a(n-3) -4*a(n-4) +a(n-5) -2*a(n-6) +a(n-7)

%F k=3: [order 19] for n>23

%F k=4: [order 77] for n>81

%e Some solutions for n=5 k=4

%e ..0..1..0..1. .0..0..0..1. .0..0..1..1. .0..1..0..0. .0..1..0..0

%e ..1..0..1..0. .1..0..0..0. .1..1..1..1. .0..0..0..0. .1..0..0..0

%e ..0..0..0..1. .1..0..0..0. .0..1..0..1. .0..0..0..0. .0..0..0..0

%e ..0..0..0..1. .0..0..0..1. .1..1..1..1. .1..1..0..0. .0..0..0..1

%e ..0..0..0..0. .0..0..0..0. .1..1..1..0. .0..0..0..1. .1..1..0..1

%Y Column 1 is A000045(n+1).

%Y Column 2 is A300500.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Mar 22 2018