A329830 Beatty sequence for (4+sqrt(10))/3.

2, 4, 7, 9, 11, 14, 16, 19, 21, 23, 26, 28, 31, 33, 35, 38, 40, 42, 45, 47, 50, 52, 54, 57, 59, 62, 64, 66, 69, 71, 74, 76, 78, 81, 83, 85, 88, 90, 93, 95, 97, 100, 102, 105, 107, 109, 112, 114, 116, 119, 121, 124, 126, 128, 131, 133, 136, 138, 140, 143, 145

Offset: 1

Author

Clark Kimberling, Dec 31 2019

Keywords

• nonn
• easy

Comments

Let r = (2+sqrt(10))/3. Then (floor(n*r)) and (floor(n*r + 2r/3)) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Links

• Eric Weisstein's World of Mathematics, Beatty Sequence.
• Index entries for sequences related to Beatty sequences

Crossrefs

Cf. A329825, A329829 (complement).

Programs

Mathematica

t = 2/3; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329829 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329830 *)

Formula

a(n) = floor(n*s), where s = (4+sqrt(10))/3.