A264166 -id:A264166 - cp's OEIS frontend

A264285 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change 0,0 0,1 1,0 or -1,-2.

1, 4, 1, 8, 10, 1, 16, 32, 26, 1, 33, 102, 132, 69, 1, 69, 360, 675, 556, 181, 1, 145, 1228, 4189, 4484, 2324, 476, 1, 300, 4156, 23852, 47492, 29742, 9724, 1252, 1, 624, 14148, 134432, 448821, 537057, 197283, 40692, 3292, 1, 1300, 48188, 768664, 4227024

Offset: 1

R. H. Hardin, Nov 10 2015

Table starts
.1.....4.......8........16..........33.............69..............145
.1....10......32.......102.........360...........1228.............4156
.1....26.....132.......675........4189..........23852...........134432
.1....69.....556......4484.......47492.........448821..........4227024
.1...181....2324.....29742......537057........8405669........131452948
.1...476....9724....197283.....6080234......157344756.......4076914388
.1..1252...40692...1308629....68815948.....2943284092.....126311779972
.1..3292..170268...8680430...778858184....55051679668....3911932445892
.1..8657..712468..57579243..8815152033..1029653214581..121136137544916
.1.22765.2981244.381936079.99770013733.19257696830753.3750881659750212

			Some solutions for n=4 k=4
..7..0..9..2..3....7..1..9..2..4....0..1..9..2..3....0..1..9..2..3
.12..1..6..8..4....0.13..6..3..8....5.13..6..7..4...12..5.14..7..4
..5.11.19.13.14....5.10.11.12.14...10.11.12..8.14...10..6.11..8.13
.10.23.16.17.18...22.15.16.18.19...22.15.24.17.18...15.23.16.17.19
.15.20.21.22.24...20.21.17.23.24...20.16.21.23.19...20.21.22.18.24

R. H. Hardin, Table of n, a(n) for n = 1..161

Column 2 is A099234(n+1).

Row 1 is A264166.

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = a(n-1) +3*a(n-2) +3*a(n-3) +a(n-4)
k=3: a(n) = 3*a(n-1) +4*a(n-2) +4*a(n-3)
k=4: a(n) = 7*a(n-1) -2*a(n-2) -2*a(n-3) -6*a(n-4) +a(n-5) +3*a(n-6)
k=5: [order 28]
k=6: [order 36]
k=7: [order 34]
Empirical for row n:
n=1: a(n) = a(n-1) +a(n-2) +2*a(n-3) +a(n-4) +a(n-5) -a(n-6)
n=2: a(n) = a(n-1) +4*a(n-2) +10*a(n-3) +12*a(n-4) +8*a(n-5) for n>7
n=3: [order 56]

A297224 T(n,k)=Number of nXk 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 1 neighboring 1s.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 8, 9, 1, 6, 16, 24, 19, 1, 9, 33, 57, 68, 41, 1, 13, 69, 182, 207, 196, 88, 1, 19, 145, 535, 997, 751, 564, 189, 1, 28, 300, 1513, 4210, 5570, 2720, 1620, 406, 1, 41, 624, 4415, 16658, 33158, 30946, 9861, 4660, 872, 1, 60, 1300, 12832, 68769, 178469

Offset: 1

R. H. Hardin, Dec 27 2017

Table starts
.1...2.....3......4.......6.........9.........13..........19............28
.1...4.....8.....16......33........69........145.........300...........624
.1...9....24.....57.....182.......535.......1513........4415.........12832
.1..19....68....207.....997......4210......16658.......68769........284867
.1..41...196....751....5570.....33158.....178469.....1051514.......6152761
.1..88...564...2720...30946....261939....1918732....16176806.....134671502
.1.189..1620...9861..171851...2063378...20599895...248421807....2936448567
.1.406..4660..35741..955316..16277793..221333623..3819208252...64142817874
.1.872.13396.129540.5308160.128351805.2377449633.58680928294.1400212345305

			Some solutions for n=5 k=4
..0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..0..0. .0..1..1..0
..0..0..1..0. .0..0..0..0. .0..1..0..0. .0..1..0..0. .0..0..0..0
..0..1..0..0. .0..1..0..0. .1..0..0..0. .1..0..0..0. .0..1..0..0
..0..1..1..0. .1..0..0..1. .0..1..1..0. .0..0..1..1. .1..0..0..0
..0..0..0..0. .0..0..1..0. .0..0..1..1. .0..0..0..0. .0..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..611

Column 2 is A078039.
Row 1 is A000930(n+1).
Row 2 is A264166.

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3)
k=3: a(n) = a(n-1) +4*a(n-2) +4*a(n-3)
k=4: a(n) = a(n-1) +6*a(n-2) +11*a(n-3) +6*a(n-4) +a(n-5)
k=5: [order 9]
k=6: [order 11] for n>13
k=7: [order 16] for n>21
Empirical for row n:
n=1: a(n) = a(n-1) +a(n-3)
n=2: a(n) = a(n-1) +a(n-2) +2*a(n-3) +a(n-4) +a(n-5) -a(n-6)
n=3: [order 13]
n=4: [order 27]
n=5: [order 60]

cp's OEIS Frontend

A264285 T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change 0,0 0,1 1,0 or -1,-2.

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