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%I #40 Oct 22 2021 11:34:41
%S 1,2,3,2,3,4,3,2,3,4,5,4,3,4,3,2,3,4,5,4,5,6,5,4,3,4,5,4,3,4,3,2,3,4,
%T 5,4,5,6,5,4,5,6,7,6,5,6,5,4,3,4,5,4,5,6,5,4,3,4,5,4,3,4,3,2,3,4,5,4,
%U 5,6,5,4,5,6,7,6,5,6,5,4,5,6,7,6,7,8,7,6,5,6,7,6,5,6,5,4,3,4,5,4,5,6
%N a(n) = 1 + Sum_{k=0..n-1} 2 * A014577(k) - 1.
%C Let s(0)=1; s(n+1)=s(n),ri(n), where ri(n) is s(n) reversed and incremented. Each s(n) is an initial part of this sequence.
%C For each m, a(1 to 2^m) is a permutation of A063787(1 to 2^m). For k=1 to 2^m, a(2^m+1-A088372(m,k)) = A063787(k).
%C Partial sums give A164910: (1, 3, 6, 8, 11, 15, 20, ...).
%C a(0) = 1, then using the dragon curve sequence A014577: (1, 1, 0, 1, 1, ...) as a code: (1 = add to current term, 0 = subtract from current term, to get the next term), see example.
%C Rows of A088696 tend to this sequence.
%H Michael De Vlieger, <a href="/A088748/b088748.txt">Table of n, a(n) for n = 0..16383</a>
%H J.-P. Allouche, G.-N. Han, and J. Shallit, <a href="https://arxiv.org/abs/2006.08909">On some conjectures of P. Barry</a>, arXiv:2006.08909 [math.NT], 2020.
%H J.-P. Allouche and J. Shallit, <a href="https://arxiv.org/abs/2006.04708">On three conjectures of P. Barry</a>, arxiv preprint arXiv:2006.04708 [math.NT], June 8 2020.
%H Paul Barry, <a href="https://arxiv.org/abs/2005.04066">Some observations on the Rueppel sequence and associated Hankel determinants</a>, arXiv:2005.04066 [math.CO], 2020.
%H Paul Barry, <a href="https://arxiv.org/abs/2104.05593">On the Gap-sum and Gap-product Sequences of Integer Sequences</a>, arXiv:2104.05593 [math.CO], 2021.
%H Paul Barry, <a href="https://arxiv.org/abs/2107.00442">Conjectures and results on some generalized Rueppel sequences</a>, arXiv:2107.00442 [math.CO], 2021.
%F a(n) = 1 + A005811(n). [_Joerg Arndt_, Dec 11 2012]
%e The first 8 terms of the sequence = (1, 2, 3, 2, 3, 4, 3, 2), where the first four terms = (1, 2, 3, 2). Reverse, add 1, getting (3, 4, 3, 2), then append.
%e The sequence begins with "1", then using the dragon curve coding, we get:
%e 1...2...3...2...3...4... = A088748
%e ....1...1...0...1...1... = A014577, the dragon curve.
%t Array[1 + Sum[2 (1 - (((Mod[#1, 2^(#2 + 2)]/2^#2)) - 1)/2) - 1 &[k, IntegerExponent[k, 2]], {k, # - 1}] &, 102] (* _Michael De Vlieger_, Aug 26 2020 *)
%Y Cf. A014577, A063787, A088208, A088372, A088696, A164910, A005811.
%K nonn
%O 0,2
%A _Gary W. Adamson_, Oct 14 2003
%E Edited by _Don Reble_, Nov 15 2005
%E Additional comments from _Gary W. Adamson_, Aug 30 2009
%E Edited by _N. J. A. Sloane_, Sep 06 2009
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