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 A165386 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, and 4 in the lower right corner. 2

%I

%S 3,40,305,1622,6868,25036,82224,250208,718536,1972132,5220384,

%T 13417184,33652952,82699012,199729888,475260768,1116459752,2593550148,

%U 5965976992,13605076608,30787314104,69191005796,154539160288,343241755840

%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, and 4 in the lower right corner.

%H R. H. Hardin, <a href="/A165386/b165386.txt">Table of n, a(n) for n=2..58</a>

%F Empirical: a(n) = 12*a(n-1) - 63*a(n-2) + 190*a(n-3) - 363*a(n-4) + 456*a(n-5) - 377*a(n-6) + 198*a(n-7) - 60*a(n-8) + 8*a(n-9) for n>=13.

%F Conjectures from _Colin Barker_, Feb 26 2018: (Start)

%F G.f.: x^2*(3 + 4*x + 14*x^2 - 88*x^3 + 108*x^4 + 8*x^5 - 98*x^6 + 60*x^7 + 9*x^8 - 16*x^9 + 4*x^10) / ((1 - x)^6*(1 - 2*x)^3).

%F a(n) = (-314880 + 315315*2^n - (139264+79785*2^n)*n + 5*(-5248+1323*2^n)*n^2 - 3280*n^3 - 160*n^4 - 16*n^5) / 240 for n>3.

%F (End)

%e Some solutions for n=4:

%e ...1.2.2.2.......1.1.2.2.......1.2.2.2.......1.1.3.2.......1.1.2.2....

%e .....2.2.2.3.......2.2.2.2.......4.4.4.4.......3.3.2.2.......3.3.2.4..

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

%e ------

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

%e .....1.3.2.2.......1.3.2.4.......3.3.3.3.......1.3.2.4.......3.3.3.3..

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

%e ------

%e ...1.2.2.2.......1.1.3.2.......1.1.2.2.......1.1.1.2.......1.1.2.2....

%e .....3.3.3.3.......3.3.3.3.......1.1.1.1.......2.2.2.2.......1.3.2.4..

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

%K nonn

%O 2,1

%A _R. H. Hardin_, Sep 17 2009

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Last modified May 28 22:08 EDT 2022. Contains 354122 sequences. (Running on oeis4.)