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A106841
Numbers m such that m, m+1 and m+2 have odd part of the form 4*k+1.
4
8, 16, 32, 40, 64, 72, 80, 104, 128, 136, 144, 160, 168, 200, 208, 232, 256, 264, 272, 288, 296, 320, 328, 336, 360, 392, 400, 416, 424, 456, 464, 488, 512, 520, 528, 544, 552, 576, 584, 592, 616, 640, 648, 656, 672, 680, 712, 720, 744, 776, 784, 800, 808
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
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
Equals 8 * A091072.
Sequence in context: A020948 A219547 A259751 * A260711 A139598 A137243
KEYWORD
nonn
AUTHOR
Ralf Stephan, May 03 2005
STATUS
approved