A329832 Beatty sequence for (9+sqrt(65))/8.

2, 4, 6, 8, 10, 12, 14, 17, 19, 21, 23, 25, 27, 29, 31, 34, 36, 38, 40, 42, 44, 46, 49, 51, 53, 55, 57, 59, 61, 63, 66, 68, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 91, 93, 95, 98, 100, 102, 104, 106, 108, 110, 113, 115, 117, 119, 121, 123, 125, 127, 130, 132

Offset: 1

Author

Clark Kimberling, Dec 31 2019

Keywords

• nonn
• easy

Comments

Let r = (7+sqrt(65))/8. Then (floor(n*r)) and (floor(n*r + r/4)) 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, A329831 (complement).

Programs

Mathematica

t = 1/4; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329831 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329832 *)

Formula

a(n) = floor(n*s), where s = (9+sqrt(65))/8.