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A188554 Number of 3 X n binary arrays without the pattern 0 1 diagonally, vertically, antidiagonally or horizontally. 2

%I #50 Feb 10 2024 20:45:38

%S 1,4,7,12,20,32,49,72,102,140,187,244,312,392,485,592,714,852,1007,

%T 1180,1372,1584,1817,2072,2350,2652,2979,3332,3712,4120,4557,5024,

%U 5522,6052,6615,7212,7844,8512,9217,9960,10742,11564,12427,13332,14280,15272,16309

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

%C 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.

%C 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

%H Alois P. Heinz, <a href="/A188554/b188554.txt">Table of n, a(n) for n = 0..10000</a> (terms n = 1..200 from R. H. Hardin)

%F 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

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

%F G.f.: -(x^4 - 4*x^3 + 3*x^2 - 1)/(x - 1)^4. - _Colin Barker_, Mar 18 2012

%e Some solutions for 3 X 3:

%e 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1

%e 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1

%e 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 1

%e 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

%Y Row 3 of A188553.

%K nonn,easy

%O 0,2

%A _R. H. Hardin_, Apr 04 2011

%E a(0)=1 prepended by _Alois P. Heinz_, Feb 10 2024

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