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

Author

R. H. Hardin, Apr 04 2011

Keywords

• nonn
• easy

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

Examples

			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
		

Links

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

Crossrefs

Row 3 of A188553.

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

Extensions

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