A045774 Extension of Beatty sequence; complement of A045775.

0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 65

Offset: 0

Author

Aviezri S. Fraenkel

Keywords

• nonn

Comments

(s,t)-sequences; the case s=3, t=2.

Links

• A. S. Fraenkel, Heap games, numeration systems and sequences, arXiv:math/9809074 [math.CO], 1998; Annals of Combinatorics, 2 (1998), 197-210.
• Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
• Wen An Liu and Xiao Zhao, Adjoining to (s,t)-Wythoff's game its P-positions as moves, Discrete Applied Mathematics, 179 (2014), 28-43. See Table 5.
• Index entries for sequences related to Beatty sequences

Crossrefs

Cf. A045749, A045750.

Programs

Mathematica

s=3; 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}] (* A045774 *)
Table[b[n],{n,200}] (* A045775 *)
(* Clark Kimberling, Apr 02 2011 *)

Formula

a(n)=mex {a(i), b(i):0 <= iA045775, mex S=least integer >= 0 not in the sequence S.