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

%I A106841 #25 Sep 14 2024 06:50:40
%S A106841 8,16,32,40,64,72,80,104,128,136,144,160,168,200,208,232,256,264,272,
%T A106841 288,296,320,328,336,360,392,400,416,424,456,464,488,512,520,528,544,
%U A106841 552,576,584,592,616,640,648,656,672,680,712,720,744,776,784,800,808
%N A106841 Numbers m such that m, m+1 and m+2 have odd part of the form 4*k+1.
%C A106841 Either of form 2a(m) or 32k + 8, k >= 0, 0 < m < n.
%C A106841 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
%C A106841 The asymptotic density of this sequence is 1/16. - _Amiram Eldar_, Sep 14 2024
%H A106841 Vincenzo Librandi, <a href="/A106841/b106841.txt">Table of n, a(n) for n = 1..1000</a>
%H A106841 Bruce Bates, Martin Bunder, and Keith Tognetti, <a href="http://pefmath.etf.rs/vol4num1/AADM-Vol4-No1-96-118.pdf">Mirroring and Interleaving in the Paperfolding Sequence</a>, Applicable Analysis and Discrete Mathematics, Volume 4, Number 1, April 2010, pages 96-118.
%e A106841 40/8 = 5 is 1 mod 4 and so is 41 and 42/2 = 21, thus 40 is in sequence.
%t A106841 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 *)
%o A106841 (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
%Y A106841 Cf. A091072, A106840, A106838.
%Y A106841 Equals 8 * A091072.
%K A106841 nonn
%O A106841 1,1
%A A106841 _Ralf Stephan_, May 03 2005