OFFSET
1,2
COMMENTS
From Andrew Howroyd, May 10 2017: (Start)
Number of n X k binary matrices with every 1 vertically or horizontally adjacent to some 0.
Number of dominating sets in the grid graph P_n X P_k. (End)
LINKS
Stephan Mertens, Table of n, a(n) for n = 1..946 (first 198 terms from R. H. Hardin)
Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], Aug 2024.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Dominating Set
Wikipedia, Dominating Set
FORMULA
Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2) +a(n-3).
k=2: a(n) = 3*a(n-1) +2*a(n-2) +2*a(n-3) -a(n-4) -a(n-5).
k=3: a(n) = 6*a(n-1) +5*a(n-2) +22*a(n-3) +7*a(n-4) +8*a(n-5) -18*a(n-6) -20*a(n-7) -a(n-8) +4*a(n-9) +3*a(n-10) +a(n-12).
Column k=1 for an underlying 0..z array: a(n) = sum(i=1..2z+1){a(n-i)} z=1,2,3,4
EXAMPLE
Table starts
....1.......3...........5..............9.................17
....3......11..........41............149................547
....5......41.........291...........2069..............14811
....9.....149........2069..........28661.............401253
...17.....547.......14811.........401253...........10982565
...31....2007......105913........5609569..........300126903
...57....7361......757305.......78394141.........8199377227
..105...27001.....5415209.....1095695529.......224032447213
..193...99043....38722037....15314367301......6121258910011
..355..363299...276885777...214044940145....167250519310183
..653.1332617..1979899795..2991651891557...4569773233045519
.1201.4888173.14157473937.41813576818545.124859601874166153
...
Some solutions for n=3 k=4
..1..0..1..1....1..1..1..0....1..1..1..0....1..0..1..1....1..0..1..1
..1..0..1..0....1..0..1..0....0..0..1..0....1..0..1..1....1..1..0..1
..0..0..1..0....1..1..0..1....0..1..1..1....1..1..1..1....1..1..1..0
CROSSREFS
Diagonal is A133515.
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Oct 26 2012
STATUS
approved