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A185526 Number of (n+2) X 3 binary arrays with each 3 X 3 subblock nonsingular. 1
174, 726, 3030, 12630, 52662, 219606, 915702, 3818262, 15921462, 66389334, 276830070, 1154326422, 4813311414, 20070547926, 83690183670, 348971383830, 1455141093942, 6067648253526, 25300883554422, 105499640297622 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,easy
AUTHOR
R. H. Hardin, Jan 30 2011
STATUS
approved

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Last modified April 13 02:05 EDT 2024. Contains 371639 sequences. (Running on oeis4.)