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Number of (n+1) X (1+1) 0..2 arrays with no element greater than all horizontal neighbors or equal to all vertical neighbors.
1

%I #13 Sep 10 2024 12:38:07

%S 6,12,36,96,264,720,1968,5376,14688,40128,109632,299520,818304,

%T 2235648,6107904,16687104,45590016,124554240,340288512,929685504,

%U 2539948032,6939267072,18958430208,51795394560,141507649536,386606088192

%N Number of (n+1) X (1+1) 0..2 arrays with no element greater than all horizontal neighbors or equal to all vertical neighbors.

%H R. H. Hardin, <a href="/A239171/b239171.txt">Table of n, a(n) for n = 1..210</a>

%H N. H. Bong, C. Dalfó, and M. À. Fiol, and D. Závacká, <a href="https://arxiv.org/abs/2409.02125">Some inner metric parameters of a digraph: Iterated line digraphs and integer sequences</a>, arXiv:2409.02125 [math.CO], 2024. See p. 19.

%F Empirical: a(n) = 2*a(n-1) + 2*a(n-2).

%F Conjectures from _Colin Barker_, Oct 25 2018: (Start)

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

%F a(n) = sqrt(3)*((1+sqrt(3))^n-(1-sqrt(3))^n).

%F (End)

%e Some solutions for n=5:

%e ..2..2....0..0....2..2....1..1....1..1....2..2....1..1....2..2....0..0....2..2

%e ..0..0....2..2....0..0....2..2....0..0....0..0....0..0....1..1....2..2....0..0

%e ..1..1....1..1....1..1....0..0....2..2....2..2....2..2....0..0....2..2....1..1

%e ..0..0....1..1....1..1....2..2....1..1....0..0....0..0....2..2....1..1....1..1

%e ..2..2....0..0....2..2....0..0....2..2....2..2....0..0....1..1....0..0....0..0

%e ..1..1....2..2....0..0....1..1....0..0....0..0....2..2....0..0....1..1....1..1

%Y Column 1 of A239178.

%K nonn

%O 1,1

%A _R. H. Hardin_, Mar 11 2014