A228661 Number of 2 X n binary arrays with top left value 1 and no two ones adjacent horizontally, diagonally or antidiagonally.

2, 2, 8, 14, 38, 80, 194, 434, 1016, 2318, 5366, 12320, 28418, 65378, 150632, 346766, 798662, 1838960, 4234946, 9751826, 22456664, 51712142, 119082134, 274218560, 631464962, 1454120642, 3348515528, 7710877454, 17756424038, 40889056400

Offset: 1

Author

R. H. Hardin, Aug 29 2013

Keywords

• nonn

Comments

Row 2 of A228660.

The recurrence is demonstrated as follows: For every 2X(n-1) array, we can add the column (0,0) to get an appropriate array of size 2Xn, and for every 2X(n-2) array, we can add the column (0,0) and either (1,0), (0,1) or (1,1) to get an appropriate sized array. Every admissible array is of one of these two forms, and these two forms do not overlap (since their last columns are different). - Tom Edgar, Aug 29 2013

Examples

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

Links

• R. H. Hardin, Table of n, a(n) for n = 1..210
• Index entries for linear recurrences with constant coefficients, signature (1,3).

Formula

a(n) = a(n-1) +3*a(n-2).

G.f.: -2*x / ( -1+x+3*x^2 ). a(n) = 2*A006130(n-1). - R. J. Mathar, Aug 29 2013

a(n) = -2/13*sqrt(13)*(-1/2*sqrt(13)+1/2)^n + 2/13*sqrt(13)*(1/2*sqrt(13)+1/2)^n. - Tom Edgar, Aug 31 2013

G.f.: Q(0)/x -1/x, where Q(k) = 1 + 3*x^2 + (2*k+3)*x - x*(2*k+1 + 3*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 05 2013