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 A262327 Number of (n+1) X (3+1) 0..1 arrays with each row and column divisible by 3, read as a binary number with top and left being the most significant bits. 1

%I #7 Dec 31 2018 06:08:25

%S 6,15,90,351,2106,10935,65610,378351,2270106,13482855,80897130,

%T 484142751,2904856506,17417978775,104507872650,626946793551,

%U 3761680761306,22569180586695,135415083520170,812482365290751,4874894191744506

%N Number of (n+1) X (3+1) 0..1 arrays with each row and column divisible by 3, read as a binary number with top and left being the most significant bits.

%H R. H. Hardin, <a href="/A262327/b262327.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 6*a(n-1) + 9*a(n-2) - 54*a(n-3).

%F Conjectures from _Colin Barker_, Dec 31 2018: (Start)

%F G.f.: 3*x*(2 - 7*x - 18*x^2) / ((1 - 3*x)*(1 + 3*x)*(1 - 6*x)).

%F a(n) = 3^(n-2)*(14 + 2^(2+n)) / 2 for n even.

%F a(n) = 3^(n-2)*(28 + 2^(2+n)) / 2 for n odd.

%F (End)

%e Some solutions for n=4:

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

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

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

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

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

%Y Column 3 of A262332.

%K nonn

%O 1,1

%A _R. H. Hardin_, Sep 18 2015

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