A184119 Upper s(n)-Wythoff sequence, where s(n) = 2n - 1; complement of A136119.

2, 6, 9, 12, 16, 19, 23, 26, 30, 33, 36, 40, 43, 47, 50, 53, 57, 60, 64, 67, 70, 74, 77, 81, 84, 88, 91, 94, 98, 101, 105, 108, 111, 115, 118, 122, 125, 129, 132, 135, 139, 142, 146, 149, 152, 156, 159, 163, 166, 170, 173, 176, 180, 183, 187, 190, 193, 197, 200, 204, 207, 210, 214, 217, 221, 224, 228, 231, 234, 238, 241, 245, 248, 251, 255, 258, 262, 265, 269, 272, 275, 279, 282, 286, 289, 292, 296, 299, 303, 306, 309, 313, 316, 320, 323, 327, 330, 333, 337, 340

Offset: 1

Author

Clark Kimberling, Jan 09 2011

Keywords

• nonn

Comments

See A184117 for the definition of lower and upper s(n)-Wythoff sequences.

(a(n)) is an inhomogeneous Beatty sequence, the complement of the inhomogeneous Beatty sequence (A136119(n)) = (floor(sqrt(2)*n + 1 - sqrt(2)/2)). See the paper by Fraenkel. - Michel Dekking, Jan 31 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..2000
• Aviezri S. Fraenkel, Iterated floor function, algebraic numbers, discrete chaos, Beatty subsequences, semigroups, Transactions of the American Mathematical Society 341.2 (1994): p. 640.

Crossrefs

Cf. A136119, A184117.

Programs

Magma

[Floor((2+Sqrt(2))*n-Sqrt(2)/2): n in [1..80]]; // Vincenzo Librandi, Jan 31 2017

Mathematica

k=2; r=1;
mex:=First[Complement[Range[1, Max[#1]+1], #1]]&;
s[n_]:=k*n-r; a[1]=1; b[n_]:=b[n]=s[n]+a[n];
a[n_]:=a[n]=mex[Flatten[Table[{a[i], b[i]},{i, 1, n-1}]]];
Table[s[n], {n, 30}]
Table[a[n], {n, 100}]
Table[b[n], {n, 100}]
Table[(Floor[(2 + Sqrt[2]) n - Sqrt[2]/2]), {n, 100}] (* Vincenzo Librandi, Jan 31 2017 *)

Formula

a(n) = floor((2+sqrt(2))*n - sqrt(2)/2). - Michel Dekking, Jan 31 2017