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%I #14 Jul 15 2022 11:31:00
%S 4,5,6,8,10,14,18,26,34,50,66,98,130,194,258,386,514,770,1026,1538,
%T 2050,3074,4098,6146,8194,12290,16386,24578,32770,49154,65538,98306,
%U 131074,196610,262146,393218,524290,786434,1048578,1572866,2097154,3145730
%N 1/4 the number of (n+1) X 4 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.
%C Column 3 of A209727.
%H R. H. Hardin, <a href="/A209722/b209722.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3).
%F Conjectures from _Colin Barker_, Jul 12 2018: (Start)
%F G.f.: x*(4 + x - 7*x^2) / ((1 - x)*(1 - 2*x^2)).
%F a(n) = 3*2^(n/2 - 1) + 2 for n even.
%F a(n) = 2^((n + 1)/2) + 2 for n odd.
%F (End)
%e Some solutions for n=4:
%e ..2..1..2..1....2..1..2..1....1..2..1..2....1..0..2..0....2..1..2..1
%e ..0..2..0..2....0..2..0..2....2..0..2..0....0..2..1..2....0..2..0..2
%e ..2..1..2..1....1..0..1..0....0..1..0..1....1..0..2..0....1..0..1..0
%e ..0..2..0..2....0..2..0..2....2..0..2..0....0..2..1..2....0..2..0..2
%e ..2..1..2..1....2..1..2..1....0..1..0..1....1..0..2..0....1..0..1..0
%Y Cf. A209727.
%Y 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 nonn
%O 1,1
%A _R. H. Hardin_, Mar 12 2012
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