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Number of (n+1) X 3 binary arrays with rows and columns in nondecreasing order and with no 2 X 2 subblock sum differing from a horizontal or vertical neighbor subblock sum by more than one.
1

%I #8 Apr 11 2018 08:46:22

%S 11,16,36,73,134,225,354,529,758,1049,1410,1849,2374,2993,3714,4545,

%T 5494,6569,7778,9129,10630,12289,14114,16113,18294,20665,23234,26009,

%U 28998,32209,35650,39329,43254,47433,51874,56585,61574,66849,72418,78289,84470

%N Number of (n+1) X 3 binary arrays with rows and columns in nondecreasing order and with no 2 X 2 subblock sum differing from a horizontal or vertical neighbor subblock sum by more than one.

%C Column 2 of A184071.

%H R. H. Hardin, <a href="/A184064/b184064.txt">Table of n, a(n) for n = 1..200</a>

%F Conjectures from _Colin Barker_, Apr 11 2018: (Start)

%F G.f.: x*(11 - 28*x + 38*x^2 - 19*x^3 + 5*x^4 - x^5 + 2*x^6) / (1 - x)^4.

%F a(n) = (4*n^3 - 15*n^2 + 74*n - 93) / 3 for n>3.

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>7.

%F (End)

%e Some solutions for 4 X 3:

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

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

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

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

%e ...

%e ...1..2.......4..4.......0..1.......1..2.......0..0.......2..3.......2..3...

%e ...2..2.......4..4.......1..2.......2..3.......0..0.......2..2.......2..2...

%e ...3..3.......4..4.......2..2.......3..3.......0..0.......3..2.......3..3...

%Y Cf. A184071.

%K nonn

%O 1,1

%A _R. H. Hardin_, Jan 09 2011