A185526 Number of (n+2) X 3 binary arrays with each 3 X 3 subblock nonsingular.

174, 726, 3030, 12630, 52662, 219606, 915702, 3818262, 15921462, 66389334, 276830070, 1154326422, 4813311414, 20070547926, 83690183670, 348971383830, 1455141093942, 6067648253526, 25300883554422, 105499640297622

Offset

1,1

Comments

Column 1 of A185534.

From Robert Israel, Dec 21 2015: (Start)

For each 2 X 3 binary matrix A, let f(n,A) be the number of such matrices with last two rows A. Then f(n+1,A) = Sum_B T(A,B) f(n,B) where T(A,B) = 1 if the first row of B is the last row of A and the matrix formed from A and the last row of B is nonsingular, otherwise 0. It can be verified explicitly that all f(n,A) satisfy the recurrence: x(n)=3*x(n-1)+2*x(n-2)+12*x(n-3) for n=4.

Therefore they all satisfy the same recurrence for all n>= 4, and then a(n) satisfies the same recurrence. (End)

Links

R. H. Hardin, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (3,2,12).

Formula

a(n) = 3*a(n-1) + 2*a(n-2) + 12*a(n-3).

G.f.: 6*x*(29+34*x+84*x^2)/(1-3*x-2*x^2-12*x^3). - Robert Israel, Dec 21 2015

Example

Some solutions for 4 X 3
..1..1..0....1..0..0....1..0..0....0..1..1....1..0..0....0..1..1....1..1..1
..0..1..0....0..1..0....0..1..1....0..1..0....1..1..0....0..0..1....1..0..0
..0..0..1....1..0..1....0..1..0....1..0..1....0..1..1....1..1..0....0..1..0
..1..0..1....0..1..1....1..0..0....0..0..1....0..0..1....1..0..1....1..1..1

Maple

f:= gfun:-rectoproc({a(n)=3*a(n-1)+2*a(n-2)+12*a(n-3), a(1) = 174,
a(2) = 726, a(3) = 3030}, a(n), remember):
seq(f(n), n=1..100); # Robert Israel, Dec 21 2015

Crossrefs

Cf. A185534.

Sequence in context: A331586 A077395 A185534 * A248458 A053173 A185525

Adjacent sequences: A185523 A185524 A185525 * A185527 A185528 A185529

Keyword

nonn,easy

Author

R. H. Hardin, Jan 30 2011

Status

approved