A188554 Number of 3 X n binary arrays without the pattern 0 1 diagonally, vertically, antidiagonally or horizontally.

1, 4, 7, 12, 20, 32, 49, 72, 102, 140, 187, 244, 312, 392, 485, 592, 714, 852, 1007, 1180, 1372, 1584, 1817, 2072, 2350, 2652, 2979, 3332, 3712, 4120, 4557, 5024, 5522, 6052, 6615, 7212, 7844, 8512, 9217, 9960, 10742, 11564, 12427, 13332, 14280, 15272, 16309

Offset

0,2

Comments

a(n) is the number of words of length n, x(1)x(2)...x(n), on the alphabet {0,1,2,3} such that, for i=2,...,n, x(i)=either x(i-1) or x(i-1)-1.

For the bijection between arrays and words, notice that the i-th column consists of 1's and then 0's, and there are x(i)=0 to 3 1's. - Miquel A. Fiol, Feb 06 2024

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..200 from R. H. Hardin)

Formula

Proved (for the number of sequences): a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (From this, the formulas below follow.) - Miquel A. Fiol, Feb 06 2024

a(n) = (1/6)*n^3 + (11/6)*n + 2 for n>=1.

G.f.: -(x^4 - 4*x^3 + 3*x^2 - 1)/(x - 1)^4. - Colin Barker, Mar 18 2012

Example

Some solutions for 3 X 3:
  1 1 0   1 1 1   1 1 1   1 1 1   1 1 1   0 0 0   1 1 1
  0 0 0   1 1 0   1 1 1   1 1 1   1 1 1   0 0 0   1 1 1
  0 0 0   0 0 0   1 0 0   0 0 0   1 1 0   0 0 0   1 1 1
For n=3, the a(3)=12 solutions are 000, 100, 110, 210, 111, 211, 221, 321, 222, 322, 332, 333. Those corresponding to the above arrays are 110, 221, 322, 222, 332, 000, 333 (as mentioned, consider the sums of the columns of each array). - Miquel A. Fiol, Feb 06 2024

Crossrefs

Row 3 of A188553.

Sequence in context: A344421 A022809 A297554 * A020732 A339891 A310793

Adjacent sequences: A188551 A188552 A188553 * A188555 A188556 A188557

Keyword

nonn,easy

Author

R. H. Hardin, Apr 04 2011

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 10 2024

Status

approved