A295918 T(n,k)=Number of n X k 0..1 arrays with each 1 adjacent to 0 or 3 king-move neighboring 1s.

2, 3, 3, 5, 6, 5, 8, 13, 13, 8, 13, 28, 39, 28, 13, 21, 60, 115, 115, 60, 21, 34, 129, 337, 467, 337, 129, 34, 55, 277, 993, 1880, 1880, 993, 277, 55, 89, 595, 2919, 7604, 10290, 7604, 2919, 595, 89, 144, 1278, 8587, 30721, 56955, 56955, 30721, 8587, 1278, 144, 233

Offset

1,1

Comments

From Kiran Ananthpur Bacche, Sep 14 2025: (Start)

T(n, k) is the number of ways to tile an (n + 1) X (k + 1) grid with 1 X 1, 2 X 2 and 3 X 3 tiles.

The bijection : A j X j tile at (x, y) position on the grid corresponds to (j - 1) X (j - 1) cells filled with 1's in the array at (x, y). (End)

Links

R. H. Hardin, Table of n, a(n) for n = 1..311

Formula

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

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

k=5: [order 38]

k=6: [order 92]

Example

Table starts:
   2    3     5      8      13        21         34          55            89
   3    6    13     28      60       129        277         595          1278
   5   13    39    115     337       993       2919        8587         25257
   8   28   115    467    1880      7604      30721      124117        501512
  13   60   337   1880   10290     56955     314044     1732883       9562608
  21  129   993   7604   56955    431844    3261576    24650278     186318117
  34  277  2919  30721  314044   3261576   33703065   348555744    3605337986
  55  595  8587 124117 1732883  24650278  348555744  4933593439   69844332764
  89 1278 25257 501512 9562608 186318117 3605337986 69844332764 1353357158724
  ...
Some solutions for n=5 k=4
..0..0..0..1. .0..0..1..0. .0..0..1..0. .0..0..0..0. .0..1..0..0
..1..0..0..0. .1..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..1
..0..0..0..1. .0..0..1..0. .1..0..0..1. .0..1..0..1. .1..0..0..0
..0..0..0..0. .1..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..1..0
..1..0..0..0. .0..0..0..1. .1..0..0..0. .0..0..0..0. .0..0..0..0

Crossrefs

Main diagonal is A295912.

Columns 1..7 are A000045(n+2), A002478(n+1), A295913, A295914, A295915, A295916, A295917.

Sequence in context: A094585 A379837 A183322 * A296834 A242642 A178041

Adjacent sequences: A295915 A295916 A295917 * A295919 A295920 A295921

Keyword

nonn,tabl

Author

R. H. Hardin, Nov 29 2017

Status

approved