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%I #25 Feb 23 2020 01:46:09
%S 0,0,1,0,2,3,1,0,4,5,6,7,2,3,1,0,8,9,10,11,12,13,14,15,4,5,6,7,2,3,1,
%T 0,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,8,9,10,11,12,13,14,
%U 15,4,5,6,7,2,3,1,0,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49
%N Concatenation of finite strings S_0, S_1, S_2, ..., where S_0 = {0} and for k >= 1, S_k is obtained from S_{k-1} by inserting the numbers 2^(k-1) through 2^k-1 after the initial 0.
%C For example, for k = 3, take S_2 = {0,2,3,1} and insert 2^2 through 2^3-1 after the 0, so that S_3 = {0,4,5,6,7,2,3,1}. The string S_k has length 2^k.
%C A self-similar fractal sequence.
%C This is the sequence g_n at the end of Section 2 of Levine's paper. The paper also continues several other sequences that are probably not in the OEIS at present.
%D L. Levine, Fractal sequences and restricted Nim, Ars Combin. 80 (2006), 113-127.
%H T. D. Noe, <a href="/A115352/b115352.txt">Table of n, a(n) for n = 0..1022</a>
%H L. Levine, <a href="https://arxiv.org/abs/math/0409408">Fractal sequences and restricted Nim</a>, arXiv:math/0409408 [math.CO], 2004.
%H L. Levine, <a href="http://math.berkeley.edu/~levine/">Home Page</a>
%F If n=2^m-1, then a(n)=0; for all other terms, write n in binary, collapse the initial segment of 1's to a single 1 and delete the first 0. For example, a(25)=a(11001)=101=5. - Lionel Levine (levine(AT)Math.Berkeley.EDU), May 04 2006
%F a(n)=0 if n=2^m-1, otherwise A054429(2^ceiling(log_2(n+1))-n-1). - _Peter Ward_, Jan 23 2020
%F a(0)=0, a(1)=0; for all other terms, write n as 2^(m+1)+k with 0 <= k < 2^(m+1), then a(n)=2^m+k if k < 2^m, otherwise a(k). - _Peter Ward_, Jan 23 2020
%e The first few strings S_0, S_1, S_2, ... are as follows:
%e 0
%e 0,1
%e 0,2,3,1
%e 0,4,5,6,7,2,3,1
%e 0,8,9,10,11,12,13,14,15,4,5,6,7,2,3,1
%t Nest[Append[#, Join[{#[[-1, 1]]}, Range[#2, 2 #2 - 1], Rest@ #[[-1]]]] & @@ {#1,Length@ #[[-1]]} &, {{0}, {0, 1}}, 5] // Flatten (* _Michael De Vlieger_, Jan 25 2020 *)
%Y See A025480 for a similar sequence.
%K nonn,tabf
%O 0,5
%A _N. J. A. Sloane_, Mar 10 2006
%E Edited by _Robert G. Wilson v_, Apr 11 2006
%E Further edited by _N. J. A. Sloane_, Jan 16 2009
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