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A228661 Number of 2 X n binary arrays with top left value 1 and no two ones adjacent horizontally, diagonally or antidiagonally.

%I A228661 #28 May 20 2023 05:12:30
%S A228661 2,2,8,14,38,80,194,434,1016,2318,5366,12320,28418,65378,150632,
%T A228661 346766,798662,1838960,4234946,9751826,22456664,51712142,119082134,
%U A228661 274218560,631464962,1454120642,3348515528,7710877454,17756424038,40889056400
%N A228661 Number of 2 X n binary arrays with top left value 1 and no two ones adjacent horizontally, diagonally or antidiagonally.
%C A228661 Row 2 of A228660.
%C A228661 The recurrence is demonstrated as follows: For every 2X(n-1) array, we can add the column (0,0) to get an appropriate array of size 2Xn, and for every 2X(n-2) array, we can add the column (0,0) and either (1,0), (0,1) or (1,1) to get an appropriate sized array. Every admissible array is of one of these two forms, and these two forms do not overlap (since their last columns are different). - _Tom Edgar_, Aug 29 2013
%H A228661 R. H. Hardin, <a href="/A228661/b228661.txt">Table of n, a(n) for n = 1..210</a>
%H A228661 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,3).
%F A228661 a(n) = a(n-1) +3*a(n-2).
%F A228661 G.f.: -2*x / ( -1+x+3*x^2 ). a(n) = 2*A006130(n-1). - _R. J. Mathar_, Aug 29 2013
%F A228661 a(n) = -2/13*sqrt(13)*(-1/2*sqrt(13)+1/2)^n + 2/13*sqrt(13)*(1/2*sqrt(13)+1/2)^n. - _Tom Edgar_, Aug 31 2013
%F A228661 G.f.: Q(0)/x -1/x, where Q(k) = 1 + 3*x^2 + (2*k+3)*x - x*(2*k+1 + 3*x)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 05 2013
%e A228661 Some solutions for n=4
%e A228661 ..1..0..1..0....1..0..0..0....1..0..0..1....1..0..0..0....1..0..1..0
%e A228661 ..1..0..1..0....1..0..0..0....1..0..0..0....0..0..1..0....0..0..0..0
%K A228661 nonn
%O A228661 1,1
%A A228661 _R. H. Hardin_, Aug 29 2013