A088748 a(n) = 1 + Sum_{k=0..n-1} 2 * A014577(k) - 1.

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, 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, 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

Offset: 0

Author

Gary W. Adamson, Oct 14 2003

Keywords

• nonn

Comments

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.

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).

Partial sums give A164910: (1, 3, 6, 8, 11, 15, 20, ...).

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.

Rows of A088696 tend to this sequence.

Examples

			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.
The sequence begins with "1", then using the dragon curve coding, we get:
1...2...3...2...3...4... = A088748
....1...1...0...1...1... = A014577, the dragon curve.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..16383
• J.-P. Allouche, G.-N. Han, and J. Shallit, On some conjectures of P. Barry, arXiv:2006.08909 [math.NT], 2020.
• J.-P. Allouche and J. Shallit, On three conjectures of P. Barry, arxiv preprint arXiv:2006.04708 [math.NT], June 8 2020.
• Paul Barry, Some observations on the Rueppel sequence and associated Hankel determinants, arXiv:2005.04066 [math.CO], 2020.
• Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
• Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.

Crossrefs

Cf. A014577, A063787, A088208, A088372, A088696, A164910, A005811.

Programs

Mathematica

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 *)

Formula

a(n) = 1 + A005811(n). [Joerg Arndt, Dec 11 2012]

Extensions

Edited by Don Reble, Nov 15 2005

Additional comments from Gary W. Adamson, Aug 30 2009

Edited by N. J. A. Sloane, Sep 06 2009