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Number of n X 2 0,1 arrays indicating 2 X 2 subblocks of some larger (n+1) X 3 binary array having a sum of one or less, with rows and columns of the latter in lexicographically nondecreasing order.
2

%I #23 Jan 24 2017 02:38:58

%S 1,3,8,18,36,66,113,183,283,421,606,848,1158,1548,2031,2621,3333,4183,

%T 5188,6366,7736,9318,11133,13203,15551,18201,21178,24508,28218,32336,

%U 36891,41913,47433,53483,60096,67306,75148,83658,92873,102831,113571,125133

%N Number of n X 2 0,1 arrays indicating 2 X 2 subblocks of some larger (n+1) X 3 binary array having a sum of one or less, with rows and columns of the latter in lexicographically nondecreasing order.

%C Also number of binary words with 3 1's and at most n 0's that do not contain the substring 101. a(2) = 8: 111, 0111, 1110, 00111, 10011, 11001, 11100, 01110. - _Alois P. Heinz_, Jul 18 2013

%H R. H. Hardin, <a href="/A227161/b227161.txt">Table of n, a(n) for n = 0..210</a>

%F Empirical: a(n) = (1/24)*n^4 + (1/12)*n^3 + (23/24)*n^2 + (11/12)*n + 1.

%F G.f.: -(1-x+x^2)^2/(x-1)^5. - _Alois P. Heinz_, Jul 18 2013

%F Binomial transform of (1 + 2x + 3x^2 + 2x^3 + x^4), i.e., of (1 + x + x^2)^2. - _Gary W. Adamson_, Jan 23 2017

%e Some solutions for n=4:

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

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

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

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

%Y Column 2 of A227165.

%Y First differences give A177787. - _Alois P. Heinz_, Jul 18 2013

%K nonn

%O 0,2

%A _R. H. Hardin_, Jul 03 2013

%E a(0) = 1 added by _Alois P. Heinz_, Jul 18 2013