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A209721 1/4 the number of (n+1) X 3 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.

%I A209721 #43 Jul 15 2022 11:31:00
%S A209721 3,4,5,7,9,13,17,25,33,49,65,97,129,193,257,385,513,769,1025,1537,
%T A209721 2049,3073,4097,6145,8193,12289,16385,24577,32769,49153,65537,98305,
%U A209721 131073,196609,262145,393217,524289,786433,1048577,1572865,2097153,3145729
%N A209721 1/4 the number of (n+1) X 3 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.
%C A209721 Column 2 of A209727.
%C A209721 From _Richard Locke Peterson_, Apr 26 2020: (Start)
%C A209721 The formula a(n) = 2*a(n-2)-1 also fits empirically. With the given initial numbers a(1)=3, a(2)=4, a(3)=5, this new formula implies the old empirical formula. (But it is not established that the old empirical formula is true, so it is not established that the new formula is true either.) Furthermore, if the initial numbers had somehow, for example, been 3,4,6 instead, the new formula no longer implies the old formula.
%C A209721 If the new formula actually is true, it follows that a(n) is the number of distinct integer triangles that can be formed with sides of length a(n-1) and a(n-2), since the greatest length the third side can have is a(n-1)+a(n-2)-1, and the least length would be a(n-1)-a(n-2)+1. (End)
%C A209721 Conjectures: a(n) = A029744(n+1)+1. Also, a(n) = positions of the zeros in A309019(n+2) - A002487(n+2). - _George Beck_, Mar 26 2022
%H A209721 R. H. Hardin, <a href="/A209721/b209721.txt">Table of n, a(n) for n = 1..210</a>
%F A209721 Empirical: a(n) = a(n-1) +2*a(n-2) -2*a(n-3).
%F A209721 Empirical g.f.: x*(3+x-5*x^2)/((1-x)*(1-2*x^2)). [_Colin Barker_, Mar 23 2012]
%e A209721 Some solutions for n=4
%e A209721 ..2..1..2....1..2..1....0..2..1....2..0..1....1..2..0....2..1..2....0..1..0
%e A209721 ..0..2..0....2..0..2....1..0..2....1..2..0....2..0..1....0..2..0....2..0..2
%e A209721 ..1..0..1....0..1..0....0..2..1....2..0..1....1..2..0....1..0..1....1..2..1
%e A209721 ..0..2..0....2..0..2....1..0..2....1..2..0....2..0..1....0..2..0....2..0..2
%e A209721 ..1..0..1....0..1..0....0..2..1....2..0..1....1..2..0....2..1..2....1..2..1
%Y A209721 Cf. A029744, A209727.
%Y A209721 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 A209721 nonn
%O A209721 1,1
%A A209721 _R. H. Hardin_, Mar 12 2012