A281469 T(n,k)=Number of nXk 0..1 arrays with no element equal to more than two of its horizontal, diagonal or antidiagonal neighbors and with new values introduced in order 0 sequentially upwards.

1, 2, 2, 4, 8, 4, 8, 17, 25, 8, 16, 37, 24, 81, 16, 32, 78, 39, 60, 264, 32, 64, 169, 57, 96, 133, 857, 64, 128, 361, 87, 130, 207, 283, 2785, 128, 256, 778, 145, 176, 278, 405, 634, 9050, 256, 512, 1673, 241, 260, 358, 534, 897, 1419, 29407, 512, 1024, 3605, 397, 406

Offset

1,2

Comments

Table starts

...1.....2....4....8....16....32....64...128...256...512..1024..2048..4096

...2.....8...17...37....78...169...361...778..1673..3605..7774.16777.36241

...4....25...24...39....57....87...145...241...397...669..1133..1909..3229

...8....81...60...96...130...176...260...406...636...996..1586..2552..4108

..16...264..133..207...278...358...472...658...974..1476..2246..3446..5334

..32...857..283..405...534...706...924..1190..1602..2290..3374..5036..7604

..64..2785..634..897..1106..1422..1887..2429..3093..4063..5629..8037.11711

.128..9050.1419.1975..2362..2858..3681..4795..6109..7699..9985.13483.18797

.256.29407.3092.4154..4901..5793..7126..9058.11794.14920.18660.23832.31694

.512.95557.6849.8884.10240.11986.14363.17511.22411.28951.36521.45125.56983

Links

R. H. Hardin, Table of n, a(n) for n = 1..481

Formula

Empirical for column k:

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

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

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

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

k=5: [order 10] for n>15

k=6: [order 11] for n>17

k=7: [order 12] for n>19

Empirical for row n:

n=1: a(n) = 2*a(n-1)

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

n=3: a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -2*a(n-4) for n>8

n=4: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -a(n-5) for n>9

n=5: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -a(n-4) +a(n-5) -a(n-6) for n>11

n=6: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -a(n-4) +a(n-6) -a(n-7) for n>12

n=7: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -a(n-4) +a(n-7) -a(n-8) for n>16

Example

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

Crossrefs

Column 1 is A000079(n-1).

Column 2 is A240478.

Row 1 is A000079(n-1).

Sequence in context: A359488 A183397 A339490 * A302623 A302415 A303182

Adjacent sequences: A281466 A281467 A281468 * A281470 A281471 A281472

Keyword

nonn,tabl

Author

R. H. Hardin, Jan 22 2017

Status

approved