A045671 Extension of Beatty sequence; complement of A045672.

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, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70

Offset

0,3

Comments

Sequence can also be characterized by a special numeration system-see above reference.

(s,t)-sequences; the case s=2, t=2.

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

Links

Table of n, a(n) for n=0..55.

Shiri Artstein-avidan, Aviezri S. Fraenkel and Vera T. Sós, A two-parameter family of an extension of Beatty sequences, Discrete Math., 308 (2008), 4578-4588. doi:10.1016/j.disc.2007.08.070

A. S. Fraenkel, Heap games, numeration systems and sequences, Annals of Combinatorics, 2 (1998), 197-210.

A. S. Fraenkel, Recent results and questions in combinatorial game complexities, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.

A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

Index entries for sequences related to Beatty sequences

Formula

a(n) = mex{a(i), b(i):0 <= i<n}, where b=A045672, mex S=least integer >= 0 not in sequence S.

a(n) = (1+sqrt(17))/4*n+O(1). - Benoit Cloitre, Apr 23 2008

Mathematica

s=2; t=2;
mex:=First[Complement[Range[1, Max[#1]+1], #1]]&;
a[0]=0; b[n_]:=b[n]=s*a[n]+t*n;
a[n_]:=a[n]=mex[Flatten[Table[{a[i], b[i]}, {i, 0, n-1}]]];
Table[a[n], {n, 200}] (* A045671 *)
Table[b[n], {n, 200}] (* A045672 *)
(* Clark Kimberling, Apr 02 2011 *)
s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {1, 1, 1, 0}}] &, {0}, 10]; (* A285671 *)
Flatten[Position[s, 0]];  (* A045672 *)
Flatten[Position[s, 1]];  (* A045671 *)
(* - Clark Kimberling, May 02 2017 *)

Crossrefs

Cf. A026366, A045672, A285671.

Sequence in context: A287527 A045749 A187570 * A276341 A098572 A001955

Adjacent sequences: A045668 A045669 A045670 * A045672 A045673 A045674

Keyword

nonn

Author

Aviezri S. Fraenkel

Status

approved