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

0,2

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.

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.

Formula

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A324389 A105500 A288569 * A323235 A086374 A322591

Adjacent sequences: A088745 A088746 A088747 * A088749 A088750 A088751

Keyword

nonn

Author

Gary W. Adamson, Oct 14 2003

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

Status

approved