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%I #38 May 02 2017 20:47:44
%S 0,1,2,3,5,6,7,9,10,11,13,14,15,16,17,19,20,21,23,24,25,27,28,29,30,
%T 31,33,34,35,37,38,39,41,42,43,44,45,47,48,49,51,52,53,55,56,57,59,60,
%U 61,63,64,65,66,67,69,70
%N Extension of Beatty sequence; complement of A045672.
%C Sequence can also be characterized by a special numeration system-see above reference.
%C (s,t)-sequences; the case s=2, t=2.
%C For n>=1, these are the positions of 1 in the fixed point of the morphism 0->11, 1->1110; see A285671. Conjecture: -1 < n*r - a(n) < 2 for n>=0, where r = (1 + sqrt(17))/4. - _Clark Kimberling_, May 02 2017
%H Shiri Artstein-avidan, Aviezri S. Fraenkel and Vera T. Sós, <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/Papers/coatp8.pdf">A two-parameter family of an extension of Beatty sequences</a>, Discrete Math., 308 (2008), 4578-4588. <a href="http://dx.doi.org/10.1016/j.disc.2007.08.070">doi:10.1016/j.disc.2007.08.070</a>
%H A. S. Fraenkel, <a href="http://arXiv.org/abs/math.CO/9809074">Heap games, numeration systems and sequences</a>, Annals of Combinatorics, 2 (1998), 197-210.
%H A. S. Fraenkel, <a href="http://dx.doi.org/10.1016/S0304-3975(00)00062-1">Recent results and questions in combinatorial game complexities</a>, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.
%H A. S. Fraenkel, <a href="http://www.emis.de/journals/INTEGERS/papers/eg6/eg6.Abstract.html">New games related to old and new sequences</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary Equations</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%F a(n) = mex{a(i), b(i):0 <= i<n}, where b=A045672, mex S=least integer >= 0 not in sequence S.
%F a(n) = (1+sqrt(17))/4*n+O(1). - _Benoit Cloitre_, Apr 23 2008
%t s=2; t=2;
%t mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
%t a[0]=0; b[n_]:=b[n]=s*a[n]+t*n;
%t a[n_]:=a[n]=mex[Flatten[Table[{a[i],b[i]},{i,0,n-1}]]];
%t Table[a[n],{n,200}] (* A045671 *)
%t Table[b[n],{n,200}] (* A045672 *)
%t (* _Clark Kimberling_, Apr 02 2011 *)
%t s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {1, 1, 1, 0}}] &, {0}, 10]; (* A285671 *)
%t Flatten[Position[s, 0]]; (* A045672 *)
%t Flatten[Position[s, 1]]; (* A045671 *)
%t (* - _Clark Kimberling_, May 02 2017 *)
%Y Cf. A026366, A045672, A285671.
%K nonn
%O 0,3
%A _Aviezri S. Fraenkel_
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