A106838 Numbers m such that m, m+1 and m+2 have odd part of the form 4*k+3.

22, 46, 54, 86, 94, 110, 118, 150, 174, 182, 190, 214, 222, 238, 246, 278, 302, 310, 342, 350, 366, 374, 382, 406, 430, 438, 446, 470, 478, 494, 502, 534, 558, 566, 598, 606, 622, 630, 662, 686, 694, 702, 726, 734, 750, 758, 766, 790, 814, 822, 854, 862

Offset: 1

Author

Ralf Stephan, May 03 2005

Keywords

• nonn

Comments

Either of form 2a(m)+2 or 32k+22, k>=0, 0

Number points of the Heighway/Harter dragon curve starting m=0 at the origin. Those m with odd part 4k+3 (A091067) are where the curve turns right. So this sequence is the first m of each run of 3 consecutive right 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 the last of each run is integers of the form 2^p*(4k+3) with p>=3. So here the first of each run is a(n) = 8*A091067(n)-2 as Ralf Stephan already noted. - Kevin Ryde, Mar 12 2020

The asymptotic density of this sequence is 1/16. - Amiram Eldar, Sep 14 2024

Examples

			22/2=11 is 3 mod 4 and so is 23 and 24/8=3, thus 22 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. A091067, A106837, A106841.

Programs

Mathematica

opm4[n_]:=Mod[n/2^IntegerExponent[n,2],4]; Flatten[Position[Partition[ Table[opm4[n],{n,1000}],3,1],{3,3,3}]] (* Harvey P. Dale, Feb 01 2014 *)

Formula

a(n) = 8*A091067(n) - 2.

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