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A045672 Extension of Beatty sequence; complement of A045671 (apart from the initial 0).

%I A045672 #45 Oct 21 2023 06:16:41
%S A045672 0,4,8,12,18,22,26,32,36,40,46,50,54,58,62,68,72,76,82,86,90,96,100,
%T A045672 104,108,112,118,122,126,132,136,140,146,150,154,158,162,168,172,176,
%U A045672 182,186,190,196,200,204,210,214,218,224,228,232,236,240,246,250
%N A045672 Extension of Beatty sequence; complement of A045671 (apart from the initial 0).
%C A045672 (s,t)-sequences; the case s=2, t=2.
%C A045672 The sequence can also be characterized by a special numeration system-see above reference.
%C A045672 For n>=1, these are the positions of 0 in the fixed point of the morphism 0->11, 1->1110; see A285671 and Mathematica program.  Conjecture: -1 < n*r - a(n) < 3 for n>=0, where r = (5 + sqrt(17))/2. - _Clark Kimberling_, May 02 2017
%H A045672 Shiri Artstein-Avidan, Aviezri S. Fraenkel and Vera T. Sos, <a href="http://dx.doi.org/10.1016/j.disc.2007.08.070">A two-parameter family of an extension of Beatty sequences</a>, Discr. Math. 308 (2008), 4578-4588; see also <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/Papers/coatp8.pdf">preprint</a>.
%H A045672 Aviezri S. Fraenkel, <a href="https://arxiv.org/abs/math/9809074">Heap games, numeration systems and sequences</a>, arXiv:math/9809074 [math.CO], 1998; Annals of Combinatorics, 2 (1998), 197-210.
%H A045672 Aviezri S. Fraenkel, <a href="https://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 A045672 Aviezri 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 A045672 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 A045672 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%F A045672 b(n)=2a(n)+2n, where a=A045671.
%t A045672 s=2; t=2;
%t A045672 mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
%t A045672 a[0]=0; b[n_]:=b[n]=s*a[n]+t*n;
%t A045672 a[n_]:=a[n]=mex[Flatten[Table[{a[i],b[i]},{i,0,n-1}]]];
%t A045672 Table[a[n],{n,200}] (* A045671 *)
%t A045672 Table[b[n],{n,200}] (* A045672 *)
%t A045672 (* _Clark Kimberling_, Apr 02 2011 *)
%t A045672 s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {1, 1, 1, 0}}] &, {0}, 10]; (* A285671 *)
%t A045672 Flatten[Position[s, 0]];  (* A045672 *)
%t A045672 Flatten[Position[s, 1]];  (* A045671 *)
%t A045672 (* - _Clark Kimberling_, May 02 2017 *)
%Y A045672 Cf. A045671, A285671.
%K A045672 nonn
%O A045672 0,2
%A A045672 _Aviezri S. Fraenkel_