A106841 Numbers m such that m, m+1 and m+2 have odd part of the form 4*k+1.

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

Author

Ralf Stephan, May 03 2005

Keywords

• nonn

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

Examples

			40/8 = 5 is 1 mod 4 and so is 41 and 42/2 = 21, thus 40 is in sequence.
		

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.

Crossrefs

Cf. A091072, A106840, A106838.

Equals 8 * A091072.

Programs

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 *)

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