A228482 T(n,k)=Number of nXk binary arrays with top left value 1 and no two ones adjacent horizontally, vertically or antidiagonally.

1, 1, 1, 2, 2, 2, 3, 4, 4, 3, 5, 9, 14, 9, 5, 8, 19, 41, 41, 19, 8, 13, 41, 127, 172, 127, 41, 13, 21, 88, 386, 728, 728, 386, 88, 21, 34, 189, 1181, 3084, 4354, 3084, 1181, 189, 34, 55, 406, 3605, 13050, 25699, 25699, 13050, 3605, 406, 55, 89, 872, 11013, 55252, 152373

Offset: 1

Author

R. H. Hardin Aug 22 2013

Keywords

• nonn
• tabl

Comments

Table starts

..1...1.....2......3.......5.........8.........13..........21............34

..1...2.....4......9......19........41.........88.........189...........406

..2...4....14.....41.....127.......386.......1181........3605.........11013

..3...9....41....172.....728......3084......13050.......55252........233875

..5..19...127....728....4354.....25699.....152373......902042.......5342712

..8..41...386...3084...25699....211588....1748684....14433982.....119188751

.13..88..1181..13050..152373...1748684...20185842...232542935....2680777055

.21.189..3605..55252..902042..14433982..232542935..3737615288...60122232373

.34.406.11013.233875.5342712.119188751.2680777055.60122232373.1349721589622

Same recurrences as A228285 except in addition this smaller one for k=5

Examples

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

Links

• R. H. Hardin, Table of n, a(n) for n = 1..1333

Crossrefs

Column 1 is A000045

Column 2 is A078039(n-1).

Cf. A228476-A228481, A228277-A228285, A226444.

Formula

Recurrences for column k:

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

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

k=3: a(n) = a(n-1) +5*a(n-2) +4*a(n-3) -a(n-5)

k=4: a(n) = a(n-1) +10*a(n-2) +15*a(n-3) +4*a(n-4) -6*a(n-5) -a(n-6) +3*a(n-7) -a(n-8)

k=5: a(n) = 3*a(n-1) +15*a(n-2) +16*a(n-3) -11*a(n-4) -20*a(n-5) +19*a(n-6) -8*a(n-7) +a(n-9)

k=6: a(n) = a(n-1) +42*a(n-2) +147*a(n-3) +70*a(n-4) -478*a(n-5) -449*a(n-6) +1199*a(n-7) +732*a(n-8) -2727*a(n-9) +659*a(n-10) +3827*a(n-11) -5776*a(n-12) +3926*a(n-13) -1152*a(n-14) -148*a(n-15) +154*a(n-16) +32*a(n-17) -29*a(n-18) -6*a(n-19) +3*a(n-20) +a(n-21)

k=7: a(n) = a(n-1) +85*a(n-2) +432*a(n-3) +192*a(n-4) -3711*a(n-5) -5096*a(n-6) +21164*a(n-7) +27340*a(n-8) -112654*a(n-9) -37244*a(n-10) +477721*a(n-11) -464722*a(n-12) -897815*a(n-13) +3102284*a(n-14) -4149918*a(n-15) +2761082*a(n-16) -138325*a(n-17) -1353257*a(n-18) +942033*a(n-19) +64683*a(n-20) -365483*a(n-21) +80904*a(n-22) +92350*a(n-23) -27097*a(n-24) -23292*a(n-25) +2585*a(n-26) +5635*a(n-27) +1405*a(n-28) -561*a(n-29) -545*a(n-30) -173*a(n-31) -14*a(n-32) +5*a(n-33) +a(n-34)