A329828 Beatty sequence for (7+sqrt(37))/6.

2, 4, 6, 8, 10, 13, 15, 17, 19, 21, 23, 26, 28, 30, 32, 34, 37, 39, 41, 43, 45, 47, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 71, 74, 76, 78, 80, 82, 85, 87, 89, 91, 93, 95, 98, 100, 102, 104, 106, 109, 111, 113, 115, 117, 119, 122, 124, 126, 128, 130, 133

Offset: 1

Author

Clark Kimberling, Dec 31 2019

Keywords

• nonn
• easy

Comments

Let r = (5+sqrt(37))/6. Then (floor(n*r)) and (floor(n*r + r/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, A329827 (complement).

Programs

Mathematica

t = 1/3; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329827 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329828 *)

Formula

a(n) = floor(n*s), where s = (7+sqrt(37))/6.