A137178 - OEIS

%I #10 Jan 23 2019 10:53:04

%S 0,1,2,1,2,1,0,1,2,1,0,1,0,1,2,2,3,2,1,2,1,2,3,3,2,3,4,4,5,5,5,5,6,5,

%T 4,5,4,5,6,6,5,6,7,7,8,8,8,8,7,8,9,9,10,10,10,10,11,11,11,11,11,11,11,

%U 11,12,11,10,11,10,11,12,12,11,12,13,13,14,14,14,14,13,14,15,15,16,16,16

%N a(n) = sum_(1..n) [S2(n)mod 2 - floor(5*S2(n)/7)mod 2], where S2(n) = binary weight of n.

%C The graph of this sequence is a special case of de Rham's fractal curve. In general, the graph of any sequence of the form a(n)=sum_(1..n) [Sk(n)mod m - floor(p*Sk(n)/q)mod m], where Sk(n) is the digit sum of n, n in k-ary notation, p,q,m integers, gives a de Rham fractal curve. The self-symmetries of all of de Rham curves are given by the monoid that describes the symmetries of the infinite binary tree or Cantor set. This so-called period-doubling monoid is a subset of the modular group.

%H John A. Pelesko, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Pelesko/pel11.html">Generalizing the Conway-Hofstadter $10,000 Sequence</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.

%H Klaus Pinn, <a href="http://arxiv.org/abs/chao-dyn/9803012">Order and Chaos in Hofstadter's Q(n) Sequence</a>, arXiv:chao-dyn/9803012, 1998.

%H Klaus Pinn, <a href="http://arxiv.org/abs/cond-mat/9808031">A Chaotic Cousin Of Conway's Recursive Sequence</a>, arXiv:cond-mat/9808031, 1998.

%t Accumulate@ Array[Mod[#2, 2] - Mod[Floor[5 #2/7], 2] & @@ {#, DigitCount[#, 2, 1]} &, 85, 0] (* _Michael De Vlieger_, Jan 23 2019 *)

%Y Cf. A005185, A010060, A115384, A135585, A135947, A135993, A004001, A004526, A004396, A037915, A135133, A135136.

%K easy,nonn

%O 0,3

%A _Ctibor O. Zizka_, Apr 04 2008, Apr 15 2008

%E Converted references to links - _R. J. Mathar_, Oct 30 2009