login
A106838
Numbers m such that m, m+1 and m+2 have odd part of the form 4*k+3.
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
1,1
COMMENTS
Either of form 2a(m)+2 or 32k+22, 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+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
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.
FORMULA
a(n) = 8*A091067(n) - 2.
EXAMPLE
22/2=11 is 3 mod 4 and so is 23 and 24/8=3, thus 22 is in sequence.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ralf Stephan, May 03 2005
STATUS
approved