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A263053 Number of (n+1) X 2 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

%I A263053 #25 Mar 17 2022 23:44:31
%S A263053 2,2,10,10,42,42,170,170,682,682,2730,2730,10922,10922,43690,43690,
%T A263053 174762,174762,699050,699050,2796202,2796202,11184810,11184810,
%U A263053 44739242,44739242,178956970,178956970,715827882,715827882,2863311530,2863311530
%N A263053 Number of (n+1) X 2 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.
%C A263053 Each row must be either 01 or 10. The two columns are therefore binary complements with sum 2^k-1, where k = n + 1 is the number of rows. If k is even then 2^k-1 is divisible by 3 and the number of solutions is 2*(2^k-1)/3. If k is odd then 2^k-1 == 1 (mod 3) and the number of solutions is (2^k-2)/3. - _Andrew Howroyd_, Feb 03 2022
%H A263053 R. H. Hardin, <a href="/A263053/b263053.txt">Table of n, a(n) for n = 1..210</a>
%F A263053 a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3).
%F A263053 From _Colin Barker_, Jan 01 2019: (Start)
%F A263053 G.f.: 2*x / ((1 - x)*(1 - 2*x)*(1 + 2*x)).
%F A263053 a(n) = 2^n - 2/3 - (-2)^n/3.
%F A263053 (End)
%F A263053 a(n) = 2*A052992(n). - _Pascal Bisson_, Feb 03 2022
%e A263053 All solutions for n=4:
%e A263053   0 1   0 1   1 0   1 0   1 0   0 1   1 0   1 0   0 1   0 1
%e A263053   0 1   0 1   0 1   1 0   0 1   1 0   1 0   0 1   1 0   1 0
%e A263053   1 0   0 1   0 1   1 0   1 0   1 0   0 1   1 0   0 1   0 1
%e A263053   0 1   1 0   0 1   0 1   1 0   1 0   1 0   0 1   1 0   0 1
%e A263053   1 0   0 1   1 0   1 0   1 0   0 1   0 1   0 1   1 0   0 1
%o A263053 (Python) [int(2**n - 2/3 -((-2)**n)/3) for n in range(1,40)] # _Pascal Bisson_, Feb 03 2022
%Y A263053 Column 1 of A263060.
%Y A263053 Cf. A052992.
%K A263053 nonn
%O A263053 1,1
%A A263053 _R. H. Hardin_, Oct 08 2015