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 A188554 Number of 3 X n binary arrays without the pattern 0 1 diagonally, vertically, antidiagonally or horizontally. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..200 from R. H. Hardin) 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 EXAMPLE 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 CROSSREFS Row 3 of A188553. Sequence in context: A344421 A022809 A297554 * A020732 A339891 A310793 Adjacent sequences: A188551 A188552 A188553 * A188555 A188556 A188557 KEYWORD nonn,easy AUTHOR R. H. Hardin, Apr 04 2011 EXTENSIONS a(0)=1 prepended by Alois P. Heinz, Feb 10 2024 STATUS approved

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Last modified April 20 10:25 EDT 2024. Contains 371819 sequences. (Running on oeis4.)