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Number of slanted 3 X n (i=1..3) X (j=i..n+i-1) 1..4 arrays with all 1s connected, all 2s connected, all 3s connected, all 4s connected, 1 in the upper left corner, 2 in the upper right corner, 3 in the lower left corner, 4 in the lower right corner, and with no element having more than 3 neighbors with the same value.
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%I #7 Mar 26 2018 09:54:20

%S 3,40,285,1382,5472,18912,59472,174568,486352,1300936,3368528,8494280,

%T 20955536,50756232,121033104,284778120,662333776,1524945352,

%U 3479910352,7878810440,17713527824,39574755848,87916721424,194311523592

%N Number of slanted 3 X n (i=1..3) X (j=i..n+i-1) 1..4 arrays with all 1s connected, all 2s connected, all 3s connected, all 4s connected, 1 in the upper left corner, 2 in the upper right corner, 3 in the lower left corner, 4 in the lower right corner, and with no element having more than 3 neighbors with the same value.

%H R. H. Hardin, <a href="/A165395/b165395.txt">Table of n, a(n) for n=2..56</a>

%F Empirical: a(n) = 10*a(n-1) - 42*a(n-2) + 96*a(n-3) - 129*a(n-4) + 102*a(n-5) - 44*a(n-6) + 8*a(n-7) for n>=12.

%F Empirical g.f.: x^2*(3 + 10*x + 11*x^2 - 76*x^3 + 169*x^4 - 270*x^5 + 321*x^6 - 216*x^7 + 88*x^8 - 8*x^9) / ((1 - x)^4*(1 - 2*x)^3). - _Colin Barker_, Mar 26 2018

%e Some solutions for n=4:

%e ...1.1.2.2.......1.3.3.2.......1.1.2.2.......1.1.2.2.......1.1.1.2....

%e .....3.3.3.3.......3.3.2.2.......1.1.1.1.......1.3.2.2.......1.3.3.3..

%e .......3.3.3.4.......3.3.4.4.......3.3.4.4.......3.3.3.4.......3.3.3.4

%e ------

%e ...1.1.3.2.......1.1.2.2.......1.3.2.2.......1.1.3.2.......1.1.1.2....

%e .....3.3.2.3.......1.2.2.3.......3.3.2.4.......3.3.2.2.......1.1.1.1..

%e .......3.3.3.4.......3.3.3.4.......3.3.4.4.......3.2.2.4.......3.4.4.4

%e ------

%e ...1.1.1.2.......1.1.1.2.......1.1.3.2.......1.1.2.2.......1.1.2.2....

%e .....1.3.2.3.......1.3.2.4.......3.3.2.2.......1.2.3.4.......3.3.3.4..

%e .......3.3.3.4.......3.2.4.4.......3.3.3.4.......3.3.4.4.......3.3.4.4

%K nonn

%O 2,1

%A _R. H. Hardin_, Sep 17 2009