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A209723 1/4 the number of (n+1) X 5 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.

%I A209723 #11 Jul 15 2022 11:31:00
%S A209723 6,7,8,10,12,16,20,28,36,52,68,100,132,196,260,388,516,772,1028,1540,
%T A209723 2052,3076,4100,6148,8196,12292,16388,24580,32772,49156,65540,98308,
%U A209723 131076,196612,262148,393220,524292,786436,1048580,1572868,2097156
%N A209723 1/4 the number of (n+1) X 5 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.
%C A209723 Column 4 of A209727.
%H A209723 R. H. Hardin, <a href="/A209723/b209723.txt">Table of n, a(n) for n = 1..210</a>
%F A209723 Empirical: a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3).
%F A209723 Conjectures from _Colin Barker_, Jul 12 2018: (Start)
%F A209723 G.f.: x*(6 + x - 11*x^2) / ((1 - x)*(1 - 2*x^2)).
%F A209723 a(n) = 3*2^(n/2 - 1) + 4 for n even.
%F A209723 a(n) = 2^((n + 1)/2) + 4 for n odd.
%F A209723 (End)
%e A209723 Some solutions for n=4:
%e A209723 ..2..1..2..0..2....0..2..0..1..0....0..1..0..1..0....0..1..0..1..0
%e A209723 ..0..2..0..1..0....2..1..2..0..2....2..0..2..0..2....2..0..2..0..2
%e A209723 ..2..1..2..0..2....0..2..0..1..0....0..1..0..1..0....0..1..0..1..0
%e A209723 ..0..2..0..1..0....2..1..2..0..2....2..0..2..0..2....2..0..2..0..2
%e A209723 ..2..1..2..0..2....0..2..0..1..0....1..2..1..2..1....0..1..0..1..0
%Y A209723 Cf. A209727.
%Y A209723 The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - _N. J. A. Sloane_, Jul 14 2022
%K A209723 nonn
%O A209723 1,1
%A A209723 _R. H. Hardin_, Mar 12 2012