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

1,1

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.

Formula

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

Prog

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

Crossrefs

Cf. A136119, A184117.

Sequence in context: A186500 A190777 A184619 * A191873 A293789 A112870

Adjacent sequences: A184116 A184117 A184118 * A184120 A184121 A184122

Keyword

nonn

Author

Clark Kimberling, Jan 09 2011

Status

approved