OFFSET
1,1
COMMENTS
Either of form 2a(m) or 32k + 8, k >= 0, 0 < m < n.
Number points of the Heighway/Harter dragon curve starting m=0 at the origin. Those m with odd part 4k+1 (A091072) are where the curve turns left. So this sequence is the first m of each run of 3 consecutive left turns. There are no runs of 4 or more since the turn at odd m alternates left and right. Bates, Bunder, and Tognetti (theorem 19 page 104), show this sequence is integers of the form 2^p*(4k+1) with p>=3. From which a(n) = 8*A091072(n) as Ralf Stephan already noted. - Kevin Ryde, Jan 28 2020
The asymptotic density of this sequence is 1/16. - Amiram Eldar, Sep 14 2024
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Bruce Bates, Martin Bunder, and Keith Tognetti, Mirroring and Interleaving in the Paperfolding Sequence, Applicable Analysis and Discrete Mathematics, Volume 4, Number 1, April 2010, pages 96-118.
EXAMPLE
40/8 = 5 is 1 mod 4 and so is 41 and 42/2 = 21, thus 40 is in sequence.
MATHEMATICA
opn[n_]:=n/2^IntegerExponent[n, 2]; Transpose[Select[Partition[Range[ 1000], 3, 1], Mod[opn/@#, 4]=={1, 1, 1}&]][[1]] (* Harvey P. Dale, May 15 2011 *)
PROG
(PARI) lista(nn) = for(k=1, nn, if(((k/2^valuation(k, 2)-1)/2)%2==0, print1(8*k, ", "))); \\ Jinyuan Wang, Jan 30 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Ralf Stephan, May 03 2005
STATUS
approved